.
Help me understand why the engine says I had superior position most of the game.

you are just not good at chess. but your oppenoent is worse. the engine will tell that a possition is beter if the engine thinhg it is better. if you want to know the real reason you should ask it to your engine. mabey give him some money...
On move 15 Black has a significant advantage because White is underdevolped, has weaknesses on the darksquares, a bad light squared bishop and problems with king safety once the queenside opens up.
I guess you see the tactical reasons for the computers assessments?
By the way, White had a perpetual at the end of the game with 41. Qf7+ and Qf8+.

Konji_Bijeli, you're still a beginner (nothing wrong with that). You must learn about development in the opening. For instance:
Keep your queen behind enemy lines unless a tactic arises, otherwise she will be attacked continually and you will lose tempi or even the piece itself.
You usually want to castle as soon as possible, so that you don't lose one of your rooks for instance.
After castling you want your rooks to be communicated as soon as possible.
Anyway, like I said, reassess the basics of Chess, it will help a lot.

After 15. Bg5, black does indeed have a moderate advantage. White played Bg5 with the threat of e5, potentially winning the knight, but this threat is easily repelled. Rather, if you look at the pawn structure, black enjoys a nice pawn on d4 that takes away the e3 and c3 squares from white and also paralyzes white's central pawns. White's pawn structure in the center and the kingside (d3-e4-f3-g2-h3) is most likely the reason why you have a -2.0 advantage. White's kingside pieces are trapped by the light squared pawns: the knight is unable to jump to e2, f3, or even h3; the bishop is not only unable to develop due to the queen on d2, but it is a bad bishop due to the white's pawns on the light squares. Black has superior development in addition to the bishop pair.

The engine lacks human understanding. It only reads numbers. It has no sense of the dangers of a position.
To use a metaphor, a computer in a Formula1 car can provide all sorts of data but you need a human driver to race the machine.
Konji_Bijeli, you're still a beginner (nothing wrong with that). You must learn about development in the opening. For instance:
Keep your queen behind enemy lines unless a tactic arises, otherwise she will be attacked continually and you will lose tempi or even the piece itself.
You usually want to castle as soon as possible, so that you don't lose one of your rooks for instance.
After castling you want your rooks to be communicated as soon as possible.
Anyway, like I said, reassess the basics of Chess, it will help a lot.
You do realize that he/she was playing as black, right?

Konji_Bijeli, you're still a beginner (nothing wrong with that). You must learn about development in the opening. For instance:
Keep your queen behind enemy lines unless a tactic arises, otherwise she will be attacked continually and you will lose tempi or even the piece itself.
You usually want to castle as soon as possible, so that you don't lose one of your rooks for instance.
After castling you want your rooks to be communicated as soon as possible.
Anyway, like I said, reassess the basics of Chess, it will help a lot.
You do realize that he/she was playing as black, right?
Aghhh!!! No, I did not!!!

So you are asking us why the engine gives its evaluations? Get off your ass and analyze the variations it reccomends.

At move 15, Black has a gigantic lead in development, superior king safety, and even a space advantage with the pawn on d4. It is enough to look at White's kingside pieces to see how terrible his development is. The pawn advance on the queenside is now a target for attack, as White does not have the development to justify this advance. ...a5 is a standard undermining tactic that Black can use to capitalize on his superior development, exploiting White's absence of minor pieces on the queenside and disconnection of the rooks. After the bishop exchange on f6, White has serious dark square weaknesses with the absence of the bishop and Black has obtained the bishop pair advantage as well. Although the material is equal, White is utterly strategically lost.
At move 15, Black has a gigantic lead in development, superior king safety, and even a space advantage with the pawn on d4. It is enough to look at White's kingside pieces to see how terrible his development is. The pawn advance on the queenside is now a target for attack, as White does not have the development to justify this advance. ...a5 is a standard undermining tactic that Black can use to capitalize on his superior development, exploiting White's absence of minor pieces on the queenside and disconnection of the rooks. After the bishop exchange on f6, White has serious dark square weaknesses with the absence of the bishop and Black has obtained the bishop pair advantage as well. Although the material is equal, White is utterly strategically lost.
Is there a way I can start to understand these kind of adventages? Some good book for a beginner or something? I always struggle with having a plan and evaluating the position. I don't know how much to expect of a beginner when evaluating a position, but just some basic things, like how to utilize superior development or stuff like that. Like that a5 push to capitalize on the absence of his minor pieces on the queenside. Maybe I'm asking too much, I don't know.
And thanks for your help, everyone!

Is there a way I can start to understand these kind of adventages? Some good book for a beginner or something? I always struggle with having a plan and evaluating the position. I don't know how much to expect of a beginner when evaluating a position, but just some basic things, like how to utilize superior development or stuff like that. Like that a5 push to capitalize on the absence of his minor pieces on the queenside. Maybe I'm asking too much, I don't know.
And thanks for your help, everyone!
Yeah, you and I are expected to be pretty much clueless on positional evaluation, but read page 5 of this article for some guidance on books to read (article is a bit dated, so there may be some newer books that are even better, but there's nothing wrong with the older books):
http://www.chesscafe.com/text/heisman06.pdf
@yeres30: 10...Nxf3+ loses to 11.Qxf3.
To the OP: if I understand your question correctly, you gained a big positional advantage because your opponent weakened his/her dark squares a lot. It must be noted that an engine can evaluate a position and point out the vast majority of tactical mistakes in a game, but it cannot offer you a human explanation of why someone is better in a given position. If you want help in that area, you should get yourself a chess teacher (or immerse yourself in studying books on positional chess).

I highly recommend Silman's "How to Reassess Your Chess". It is a very good positional book with clear, logical explanations and many diagrams. There are also quite a few chess classics that are available for free (I'd you know where to look; "My System" by Nimzowitsch, "Chess Fundamentals" by Capablanca, and "The Art of the Middlegame" by Keres and Kotov are all very good books.

At move 15, Black has a gigantic lead in development, superior king safety, and even a space advantage with the pawn on d4. It is enough to look at White's kingside pieces to see how terrible his development is. The pawn advance on the queenside is now a target for attack, as White does not have the development to justify this advance. ...a5 is a standard undermining tactic that Black can use to capitalize on his superior development, exploiting White's absence of minor pieces on the queenside and disconnection of the rooks. After the bishop exchange on f6, White has serious dark square weaknesses with the absence of the bishop and Black has obtained the bishop pair advantage as well. Although the material is equal, White is utterly strategically lost.
Is there a way I can start to understand these kind of adventages? Some good book for a beginner or something? I always struggle with having a plan and evaluating the position. I don't know how much to expect of a beginner when evaluating a position, but just some basic things, like how to utilize superior development or stuff like that. Like that a5 push to capitalize on the absence of his minor pieces on the queenside. Maybe I'm asking too much, I don't know.
And thanks for your help, everyone!
Logical Chess: Move by Move by Irving Chernev is a book that was very helpful for me when I started as a scholastic player. The annotations are not too difficult for the beginner to follow and there are a lot of instructive master games illustrating positional play.
For a more systematic organization of positional themes, the recent book Chess Training for Post-Beginners: A Basic Course in Positional Understanding by Yaroslav Srokovski looks very relevant. It seems to be a more advanced book than Logical Chess, but it has many examples of common positional themes laid out in chapters as follows:
Pieces cut off from the play
Open files
Strong and weak squares
Weak complexes and weak diagonals
Pawn majority on the queenside
The strength of the pawn
Weak pawns
The king in the middle
Good knight versus bad bishop
Good bishop versus bad knight
The advantage of the bishop pair (in endgame and middlegame)
I haven't read enough to give a detailed review, but it looks promising and you might find this book useful to improve your positional understanding.

Konji_Bijeli, you're still a beginner (nothing wrong with that). You must learn about development in the opening. For instance:
Keep your queen unless a tactic arises, otherwise she will be attacked continually and you will lose tempi or even the piece itself.
You usually want to castle as soon as possible, so that you don't lose one of your rooks for instance.
After castling you want your rooks to be communicated as soon as possible.
Anyway, like I said, reassess the basics of Chess, it will help a lot.
I want to address these points individually:
- The queen rule is good usually, but in positions such as 1.e4,c5 2.Nf3,e6 3.d4,cxd4 4.Nxd4,Qb6!? 5.Nb3,Qc7!? the queen even moves twice! I don't even have to mention the Scandinavian but this game from white's side is an egregious violation of the rule, yet it was broken within the context of a strategy:
2."You usually want to castle as soon as possible"
There are cases where premature castling is a bad idea. Consider the following:
Again rules are good to follow, but know when to spot exceptions too.
Thanks, I'll check it out!
Thanks for the recommendations. I've heard of How to Reassess Your Chess and My System, but everyone says those are too advanced for beginners. Do you think I could get something out of them even now? What do you think about Seirawan's Winning Chess Strategies?
I'm just going through Logical Chess: Move by Move and it's a great book.
Chess Training for Post-Beginners: A Basic Course in Positional Understanding seems good judging by the chapters. I'll definitelly check it out! Thanks!

Exactly TheGreatOogieBoogie, that's why they are not rules, but principles or guidelines.
http://home.comcast.net/~danheisman/Articles/Dan_sayings.html
http://www.chesscafe.com/text/heisman91.pdf
First of all, I must mention I analyzed the game myself, and then with engine. I only later read to post before engine analysis. I hope it's no problem.
I played a game (I was black) and I thought we were equal most of the game at best (even later when I analyzed it myself), until he blundered his rook. It doesn't matter that he blundered it, nor does it matter that I won, I just want you to help me understand what makes my position that much better, because I was really surprised when I turned on the engine.
The engine says that after 9. h3, my advantage never goes under -1.00. OK, I missed a tactic that could have won me a pawn so I guess that's why, but even after all the exchange in the center, after 15. Bg5, the engine gives me approximately -2.00. Why? I can see that my development is better, but I though his pawns gave him spatial advantage, especially after 18. g4, but at that point, the engine gives me -2.87. Around there he blundered his rook.
Even though, I felt like he still had some good attack going on the kingside, which obviously wasn't the case. What makes his attack so weak that I always have -6.00 or more?
[Event "Live Chess"]
[Site "Chess.com"]
[Date "2014.08.28"]
[White "ujaladi"]
[Black "Konji_Bijeli"]
[Result "0-1"]
[WhiteElo "1054"]
[BlackElo "1182"]
[TimeControl "30|0"]
[Termination "Konji_Bijeli won by checkmate"]
1.e4 e5
2.Qf3 Nc6
3.Nc3 Nf6
4.d3 Nd4 {I wanted to gain tempo on his queen and centralize my knight. Maybe it would be better to just develop, but I guess this isn't bad.}
5.Qd1 Bb4
6.a3 Ba5
7.b4 Bb6
8.Be3 d6
9.h3 O-O
10.f3 Be6 {Missed Nxc2+, winning a pawn.}
11.Nce2 d5
12.c3 Nxe2
13.Qxe2 d4
14.cxd4 exd4
15.Bg5 {I thought we were equal here at most since I felt like white had spatial advantage with his pawns, but the engine gives -2.00.}
15...h6
16.Bxf6 Qxf6
17.h4 Bd7 {I wanted to free the path for my queen so I could play 18... Qc6 and later 19... Qc3+ winning his rook, what actually happened in the game. But better move here was 17... a5, so I could later give check with bishop, clear my way for the rook with Bc3 and winning the a-pawn. But I would never have seen it.}
18.g4 Qc6
19.g5 Qc3+
20.Kf2 Qxa1
21.gxh6 gxh6
22.f4 Qxa3
23.Nf3 Kh8
24.Ne5 Be6
25.Qh5 Kh7
26.Rg1 Rg8
27.Rxg8 Rxg8
28.f5 {The engine gives me -11.50 advantage, which seemed crazy to me }
28...f6 {Apparently, this was a mistake, and a huge one, going from -12.50 to -2.50, and the best move was Qc1, threatening Qf4+, winning a piece, probably also adding my queen to defence, and using my rook on g8 to limit white's king's movement, but that's just my guess. Probably isn't far from the truth. OK, it also threatens mate in 1 (Qe3#) if he takes my bishop. Yeah, just a really bad miss on my part, I guess.}
29.Ng4 Rxg4 {Giving up an exchange to neutralize his attack.}
30.Qxg4 Bf7
31.Be2 Qxb4
32.Qg1 a5 {And finally the a5 push.}
33.Qd1 a4
34.Bh5 Bxh5
35.Qxh5 Qd2+
36.Kg3 Qxd3+
37.Kf2 Qd2+
38.Kf1 Qd3+
39.Kf2 Qe3+
40.Kf1 d3
41.Qd1 Qf2# 0-1
How to make these moves into a board?