Simply, you must find the best move or sequence of moves in the position. Sometimes it will be something that forces checkmate, sometimes something that wins material. Usually after solving the tactic the final position will be winning for the side you are playing moves for.
Purpose of Chess Tactics?

Generally, you should end up with at least 2 pawn advantage. So, count the material before you solve a puzzle.
So, I'm a very new chess player and I recently tried out the "Tactics Training" section. I'm slightly confused as to what the win conditions of the problems are. There are situations where checking a king will solve it, but then the king could simply get out of the check in the next move, yet the problem is considered resolved. There are other problems where the king isn't checked at all and yet the problem is considered resolved.
Could someone please explain to me what the exact win condition of tactics training is?