strange pawn attack?

Hello, I just played a game and my black d5 (d-five!) pawn was taken by white e5 pawn (e-five!!!!) moving by diagonal to an empty e6! Is it a system bug or I missed some basic chess rule during my lessons?

You missed a rule, but a lot of people miss it, especially if they're taught the basic moves by a casual player (who likely also didn't know it).  It's called En Passant.

Basically, if you have a pawn on your 5th rank and your opponent moves their adjacent pawn 2 spaces as its first move in an attempt to avoid capture, you can still capture it "in passing," but only if that is your next move.  Once you make a different move, En Passant is no longer available.

check it in the rule book. It's very good documented