What Is this opening?

I was playing black in this match earlier, and being a novice player, I was unfamiliar with the opening move. I have heard it is the Van't Kruij's, but I have no idea what its purpose is. Now, the game does not follow it at all, but it still snagged my interest.

When I say new, I mean I haven't studied it. I have actually only "studied" two or three white openings. e3 isn't one of them. 

I understand that for this game it makes no difference. Bringing out the queen so early was a mistake. However, in general, I am wondering how the line is supposed to run, and how it works. 

I have seen somewhere that the line does begin 1e3 e5, but I would like to know what should follow that, (probably not 2. Qf3) and what that leads to. 

It's called The Hung Q. 

Officially it looks like a Queen's Gambit Accepted to me.

At a high level, I would imagine White to play 2. d4 and later, with or without the exchange on d4, c4 which makes it a form of the Queen's Gambit.  Your opponent tried Qf3 which indicates an attack on f7. You defended that accurately and from then on, I have no idea what your opponent thought he was doing.

White usually plays d4 on move #2 in this position usually transforms into the basic idea of the queens gambit, In which white pressures blacks c-file.

As the black side, there is no problem with capturing (2. exd4) Just continue by taking control of the center by developing 3.Nf6 which will allow you to play 4.d5, which will also take control of the center. Also keep the move c5 at the tip of your fingers for when your board has this position. (d4-d5,c4,... then c5. Just make sure to develop and protect your C-file and you should be fine.

SMTL

To answer your original question, I do believe that this opening has created a whole new realm of theoretical novelties.  What was your opponent's name?  Perhaps this opening should be named after him.

This opening is called the Patzer-WarriorDude01 attack. It's theoretically unsound because one withdraws the queen from the huddle.

During my earliest chess days, most games against my school friends started 1.e4 e5 2.Qh5, and a large number of games continued 2...g6? 3.Qxe4+,picking up the Rook in the corner.  I had the unfortunate tendency to play this with both colours.

I then realized, in a flash of insight, if I played 1.e4 e6! 2.Qh5, then I could play g6 and not lose a pawn.  In addition, the e6-pawn stops the Scholar's mate, and it prevents annoying Knights from hopping to d5.  e6 is a perfect move!

Well, if it's good for Black, it must be even better for White, and thus I spent a good 3-4 months playing 1.e3 e5 2.Qh5, and winning the vast majority of those games.  When I stopped playing kids my own ages, I learned strategies that didn't involve Scholar's mate, and thus ended the great 1.e3 period of my chess history.

1 e4 e5 2 Qf3 - Chess Kings' huge "Gigaking" database about 200 games that began this where Black appears to win about 3/4 or more of the time.  Chessgames.com's database found 4 out of about 111,000 1 e4 e5 games that continued 2 Qf3 where Black won 2 with 1 draw.

This opening is simply a "Kings Pawn Game" ECO C20, which simply means it began 1 e4 e5, then got strange. There's no traditional opening designation for it because 2 Qf3 is an awful move.  Theoretically, if White played 3 Qd1 then Black could play the Giuoco Piano, Ruy Lopez, Four Knights, etc. in reverse with a move in hand but White doesn't lose the game because of 2 Qf3.

But anyone who plays 2 Qf3 is hoping Black won't defend the f7 square and that after another move like Bc4 he can play Qxf7 mate.  So 2 Qf3 tends to be played by those who don't know much about playing chess and hope they can play a quick trap.  His third move, 3 Qf5 demonstrates he doesn't have a clue even before he hangs the Q on the next move.

So, because of the ridiculous Queen Hunt your opponent kindly provided for you, your moves don't fit any normal opening pattern.  So, as far as a "proper continuation" you asked for goes, it's hard to describe - especially after he hangs another piece on move 10.

In general, after I was a Q up, I'd have pushed the d-pawn to d5 to get my King's Bishop active and castle then, playing safely and making sure my pieces were overprotected, I slowly pick him apart. Outside of move 15, where you could have simply taken the g-pawn or moved toward the middle but you let you bishop get snagged by a well-known pawn trap, your decision to attack in the middle and your setting-up of the mating net is impressive.