1.e4 c5 2.b4!?

I was shocked by this move. I think it must not be so good as I`ve never seen it before, nor played by good nor by amateur players. However, after a few moves, seems that whit is perfeclty fine, despite being a pawn down. How could I have improved my play? Did you know this line?

which gambit is this ?? halloween ??

No, halloween is a 1.e4 gambit

1.e4 c5 2.b4 d5 I believe is supposed to do the trick

That's the wing gambit of the Sicilian. Taking the pawn is fine even though this is better:

Hi! Let's make great friends!!

As pointed out, Black's best defence is to counter with d5 (either directly on the 2nd move, declining the gambit, or after accepting it with 2...cxb4 3.a3 d5!). Therefore many who play the Sicilian Wing Gambit defer it one move by 2.Nf3, only playing b4 move 3, including against 2...Nc6 (3.b4, known as the Portsmouth Gambit), where the threat of b5 means that Black is best off capturing on b4 - with the pawn and we're back in normal Wing Gambit territory (4.a3 etc.), or by 3...Nxb4 where White reckons that the knight is displaced and hopes to capitalize on this. On the whole the Wing Gambit (just like the Morra Gambit: 1.e4 c5 2.d4 cxd4 3.c3) isn't entirely sound, but it's a dangerous weapon, especially if Black's unprepared. Paul Keres won some famous games in this gambit.

b4 is a slightly dubious move. The problem is that it gives black drawing chances

2. c3 is much more accurate

But no doubt considered much duller by those who play the Wing Gambit!

The first move that came to mind is 2...b6, but not sure how sound is it or not.

B4 is bad. 1. E4 c5 2. B4 cb 3. d4/a3 d5!

This is called the Wing Gambit. I encountered this move twice in tournament play. Once in the Oregon Open and another at my local club. I won both games as Black when I played the Sicilian defense. If White doesn't know the theory, Black can win the Queenside Rook. This line is covered well in Dorian Rogozenk's book "Anti-Socilians: A Guide for Black.

This is the famous Wing Gambit first played by Greco 400 years ago :

http://www.chessgames.com/perl/chessgame?gid=1271591

http://www.chessgames.com/perl/chessgame?gid=1271592

To me, anything that isn’t Alapin or Grand Prix.