Use a FEN string. For example:

pppppppp/ppppppp/pppppppp/pppppppp/PPPPPPPP/PPPPPPPP/PPPPPPPP/PPPPPPPP w - - 0 1. This is what the position looks like.

Post your own illegal positions here!

r3kb1r/pN3ppp/3p1q2/2pNp3/QPB1Pn2/3P1nPb/P1P2P1P/R1B3KR w kq - 0

dnleary and SerbianChessStar - Nice "find what's wrong with this position".

Stalemate:

One of those positions seems possible.  That black bishop might not have been a charter member, but might instead have been the result of a promotion.  I mean, assuming that anyone would actually promote to a bishop!

 Very nice! i hadnt thought of that

Yes, but none of the white pawns have moved, but the white rook has jumped over them. So that last position of SerbianChessStar's is illegal.

Another illegal checkmate:

 and my second one was illegal, my first one was the only legal one/.

The second position is illegal because white doesn't have a king!

I knew somebody will point this out. Since king isn't necessary to give a checkmate (except opponent's king), I didn't include him in any of the positions I posted. Just imagine that white's king is on a8 in all of the positions if you must.