what is the idea behind 3 . Ne4 in Queen Pawn opening

pl explain

There is no normal queen pawn opening with this move. 

1.d4 Nf6 2.c4 e5 3.dxe5 Ne4

That would be written as 3. ...Ne4

In the Budapest gambit black gets active piece play while white gets material a big pawn center.

Beyond that it's not really an opening you can understand through general ideas, you need to look at concrete variations.

Personally all I remember is to play a3 on the next move. 

sorry I ment 3......Ne4

If you mean 1.d4 Nf6 2.c4 e5 3.dxe5 Ne4? this is called the Fajarowicz gambit. Its main idea is expecting white to play nonsense, else Black is toast.

I have met it OTB just once, against a late friend of mine. Can't recall the whole game anymore, but it started like that:

That is the simplest way to refute such stupid gambits: Returning back the material to expose the deficiencies in the opponents' setup.

Here, black is lagging in development and central control, and white went on winning without much difficulty.

Isn't there also another opening with ...Ne4 called the Vulture?  I cannot recall if it is 1.d4 Nf6 2.c4 Ne4 or if there is a move that Black plays first and only if 3.Nf3 does he play 3...Ne4, not sure which it is.  It is unsound either way if you ask me, but I recall seeing an old book from the 80s or 90s called "The Vulture" where I believe this move was played.

"Der Geier", named by the author FM Stefan Buecker, which means "the vulture" in English.

It goes 1.d4 c5 2.d5 Nf6 3.c4 Ne4?!

I even tried this once against my friend WGM Botsari, and lost badly in less than 25 moves.

Thanks pfren, so it was c5 that I was missing.  Yeah, I can just imagine why you lost so bad with it.  Might even be worse than the Fajarowicz, and yes, your game with 4.a3 I have always viewed as the bust.  Black seems to be hoping for a move like 4.Nc3 or some other inferior move.

I'm not even 100% convinced that the Budapest (3...Ng4) is fully sound either, but it can't be nearly as bad as the Fajarowicz or Vulture.

thanks to all and all their help

A recent trend in the Budapest proper is 4.e3 Nxe5 5.f4.

At first looks, this is a poor version of Spassky's variation (where the pawn is at e4). However, keeping the pawn at e3 does have certain advantages, and in practice (mostly computer and CC games) white is scoring massively. To be honest I did not care to analyse it seriously, as I rarely play 1.d4 anymore.

I had no idea Ng4 was better than Ne4. I've never played the Budapest from either side haha.

I've played the booty proper once over the board and once via 4.Bf4 g5.  I recall drawing the first and losing the second.

Not an opening I ever have any intention of playing.  Never have or will play the Fajarowicz or Vulture.

Some big names also tried 1. d4 Nf6 2. c4 e6 3. Nf3 Ne4 almost a century ago...