that was very positional play on your part. you didnt play to recklessly but still very aggresive with a touch of risk aversion. I did not spend time analyzing the game in each move but just noticed your style of play was aggressive yet postitional...very like myself. Players that are postional usually dont go for tactical attacks but simply lock up key squares and wait to counter relatively safely.
How to go about calculating complex tactical positions?

Argh... just looked at it again.
30. Bxd6 instead of 30. Rxd6 would have been the correct move and I would've been up much more. Wasn't even a missed combination, just a single move within the combination that was completely wrong. -_-
So it indeed was a massive blunder afterall.
@ptd570: The positional play mostly comes from reading Silman's "Amateur's Mind". Great book about strategical play! But I kinda feel that my tactical ability lacks far behind.

Now that I look at it again...
Bxd6 wouldn't have been better at all!
Actually it would've been worse because:
Rxe6 would have gained the Knight instead of the bishop but also would have pinned the Bishop. But this shows well where my problem lies.

I now think that 28. Bc5 would have been the best answer to 27. ... Ta7.
He could not have attacked the pinned rook a third time but instead would've been forced to exchange it off.
@Saluang-Tewei: This is not what actually happened in the game. Everything starting from move 55. in your diagram is not the actual game anymore. Infact it only had 59 moves and was not played out to the checkmate. oO

Xilmi: Your question is similar to "How can I bench press 250 lbs?"
You start where you are and keep working to improve.
It's a matter of practice. You accumulate tactical patterns, you get better at visualizing one, two, three and more moves ahead accurately, you make sure to examine checks, captures, and other forcing moves, you learn not to overlook unexpected resources for yourself and your opponent, etc.
There is no key insight or perfect method. The people who are good at tactics are those who have worked hard at them over time. Speaking for myself, the best way I've found to improve my calculation is to solve tactical puzzles everyday.
I wouldn't ignore the engines either. Tactics are what engines do best. You can go over your own games afterward and still overlook what you missed during play. The engines won't.
Did you know you missed winning your opponent's queen at move 21?
21.Rd7 Qb6 22. Bc5.

Wow, ouch. 8[
Of course I didn't know that!
I know that good tactical opportunities like that naturally emerge from a superior position.
The problem seems to actually realize them.
I thought there probably might be some straight-forward algorithms to calculate them real quick. Like there's tricks for calculating Squareroots and multiples quickly in the head.

Your opponent let himself get backed up into his back ranks, so the opportunity developed. Sometimes though your opponent might have a fine position but nonetheless make a big mistake and you have to be ready to pounce.
In a famous game Kramnik forgot to check his king's safety properly and let Deep Fritz win with a mate-in-one.
For the most part humans don't have algorithms for playing chess, so much as they develop pattern-matchers. One such pattern as in your game is noticing when a major piece is running low on squares to move to and if so, checking if that piece can be trapped.
I've gotten in the habit of making quick sweeps through all the legal moves for both sides to see if that doesn't trigger one of my pattern-matchers. I won a recent game by looking at a nonsensical rook capture of a protected pawn but then noticing a queen fork follow-up.
There are some algortihms like counting how many times a piece is attacked and defended or visualizing a square to see if a king can catch a passed pawn, but mostly it's just patterns.

Move 26: Calculate the most forcing move first. 26. Nxd8 Rxa5 (forced) 27. Nxc6 Rxa3 (27...Nxc6 28. Bxd6) 28. Nxe7+, you're up a piece.
Move 28: It helps that you're already up a piece here; try to find ways to liquidate and exchange off pieces. 28. Rxa7 Qxa7 29. Qc5/Bb4 is simple enough; anything better? 28. Rxe7! is the sort of exchange "sac" you should always see, in this case not even a sac, winning another piece. 28. Rxe7! Qxd6 (28...Rexe7? 29. Qxb8+, overloading; 28...Raxe7 29. Qxb8 Rxb8 30. Bxe7, overloading) 29. Bxd6 (29. Rxe8+ Bxe8 30. Bxd6 Ra6 31. Nc7! also works) Raxe7 30. Bxe7 Rxe7 31. Rxc6, up 2 pieces.
Here the original 28. Rxe7 was motivated by: 3 pieces attacking it, 2 defending; Re8 overloaded defending e7 and b8; removing the defender of c6; and importantly the intuition of always looking for tactics starting by taking minor pieces with rooks.
For calculating this sort of thing, what happens in my head is that I look at the most forcing replies first, calculating if they're good enough to be played. Then I look at any sort of capture; if I'm ahead in position or material, the exchange "sac" is usually available as part of a tactical blow. Calculate those next. All the while looking for zwischenzugs, counterattacks and any sort of trick that might be available for either side.

After 27... Ra7, you can just trade rooks and remain up a piece (perhaps not best but still very solid). I just calculate all checks and captures and eliminate the blatantly bad variations (I would look about 4 moves deep if it is very tactical). If none of the capture/checks look appealing, then look at threats like unstoppable checkmate or a future knight fork. Solving tons of the 1200-2000 tactics should help with the basics. At 2500+, it becomes more about deep calculation, unstoppable threats, and weird sacrifices.
In this game:
http://www.chess.com/livechess/game?id=725227408
Fast-forward to move 26.
I felt really uncertain about how the tactical situation played out.
I am not sure if I blundered or if there was a better way for me to play it, if my opponent played the best possible or if he blundered.
It simply was too complex for me!
I kinda felt bad for losing the bishop at the end. But I don't think that I could have prevented it after taking the Knight.
I mean in the end I won quite a bit in that exchange but it's not that I could have calculated that outcome beforehand.
Can you calculate it? Are you able to come up with a better result?
I mean I know, I could simply feed it to an engine to see how that situation would ideally have turned out, but I want to learn how to calculate something like that for myself.