Before examining some complications, let's focus for a moment on one type of simple queen fork that often arises early in a game: the move of a queen from its original position to the side of the board, where it may be able to check the enemy king and also attack a loose piece in the center. First we will look at a couple of elementary examples to illustrate the idea; then we will examine what the pattern can look like from a defensive standpoint just before it arises.
In this first example, examine the protection each Black piece enjoys and notice that both rooks and (above all) the knight are unguarded. Now ask what checks White’s queen can administer and what else it can do at the same time. You see that it can attack Black's king by moving to a4, and that from there it also attacks and wins the loose knight.
This is a classic pattern. A queen from its starting position on the board often can check the enemy king by jumping to the a-file or the h-file, so long as the pawns blocking its path—first its path to the side of the board, and then its path to the enemy king—have been moved out of the way. The key squares to monitor are the c2/d7 pair and the e2/f7 pair for White; if either of those sets of squares have been evacuated by their pawns, the White queen may be able to check the enemy king with one move and simultaneously attack a loose piece near the center. Likewise the c7/d2 and e7/f2 pairs from Black’s standpoint. When you see these pairs of squares open up early while the enemy king still is in the center, think hard about any moves or exchanges that would leave a piece unprotected in the middle of the board, or that would leave a piece there attacked once and protected once.