Pinned protector
White has two pawns, one positioned to protect the other. But black's bishop has pinned the pawn at f2. If black's queen takes the pawn at g3, white cannot use the f2 pawn to take back, because doing so would expose the king to a check from black's bishop.

If it is white's move, he can protect against this attack by moving his king from behind the f2 pawn, for example, by advancing to g2, where he could protect both pawns, or moving to f1, protecting the f2 pawn, which, freed from the pin, would protect the pawn at g3.

 
Protected attacker
White can checkmate with his queen at g5 because his knight will protect his queen from being taken by black's king. This can be a very effective method of checkmating, because only one piece is needed to cover all of the squares to which the king might move.

Ignoring black's pawn for the moment, black could protect himself from checkmate by moving to h6. Then, when white moved his queen to g5, he could retreat to at h7. If black had a piece other than the king protecting g5, that would also save him from this attack. (Note that moving to h4 would just postpone checkmate, because after moving to h3, the white queen would checkmate at g3).

Counter-attacking
Often when faced with a potential check, or when a piece other than the king is being attacked, it is more effective to defend by counter-attacking than by moving to safety. For example, if black promotes his pawn to a queen at h1, he checks white's king in the process. After white moves his king to safety, black's new queen takes white's knight, and material is even.

If you can counter-attack by checking or by threatening a piece that is more valuable than your piece which is under attack, you may be able to buy time to protect yourself in a way more to your advantage than simply moving a piece to safety.
 
Trapping
Black's king is trapped behind his wall of pawns. If white moves his rook to a8, the king has nowhere to run. In this particular case, black will postpone checkmate for one turn by moving his bishop to c8, but since the bishop won't have anyone to protect him, the rook will capture it on the next turn and the game will be over.

If black had another piece that could move to c8 or a piece that he could move to f8, he would be safe because the blocking piece would be protected from white's rook. Also, if black's knight were on the a8-h1 diagonal, he would be safe because white would not be able to move safely into the checking position at a8.

If it is black's turn, he can avoid checkmate by advancing a pawn so that he can move in behind it after the rook checks, or by moving his king to f8, allowing him to escape to e7.