Black has seven legal moves in the position below, and exactly one of them wins a rook.
g Every one of those seven moves is legal, and a beginner in a hurry plays Kf8 because the king is the piece being shouted at. Then the rook goes home and the game is level again. Take the free rook.
Capture, block, move, in that order
Look for the capture first. A checking piece has walked right up to your king, which usually means it has walked away from its friends, and the whole point of a check is that your opponent gets no time to rescue it. Ask what attacks it: pawns first, then knights and bishops, then the heavy pieces, then the king itself if the checker is undefended.
Look for the block second. Count the squares between the checking piece and your king (two here: e6 and e7), and ask what can reach any of them. Blocking is the answer that lets you keep developing, and it is the answer that wins the piece back later when the block also attacks something.
Move the king last, because it is the answer that never wins anything and often costs you the right to castle. When you do move it, look at the square you are going to rather than the square you are leaving. Every year I watch someone escape a rook check by stepping onto a diagonal a bishop has been eyeing since move six.
The checks with only one answer
Two kinds of check delete the first two options at a stroke.
A knight check can never be blocked. There is no square between a knight and the piece it attacks, so your only replies are to take the knight or run.
A contact check, where the checking piece is standing next to your king, cannot be blocked either, for the same reason. And if it is defended, you cannot take it.
7 
b g Double check does the same job with the third option instead: when two pieces give check at once, no single capture and no single block can stop both, so the king has to move. That is why double check puzzles look so violent; the attacker can hang anything they like.
When all three answers are gone, the position has a different name, and it is checkmate. If you find yourself reaching for a fourth option, castling out of check is the one everybody tries, and the rules do not allow it.