A and B each represent a distinct set of commands. If the return code from set A is 0 (successful), set B will be executed. Otherwise, set B will not be executed.
For example,
#!/bin/sh FILE="/usr/tmp/junk" test -f $FILE && rm $FILEfirst verifies that the file exists, then erases it. If it did not exist, the rm command would not have been issued.
A series of commands could have been executed, as shown in the previous ``||" example.