For example, lets say you want Ctrl + C to quit the program (bare with me, very simple example), and you have a select() call. So you set up your signal handlers, and then start filling the required structures for the select() call, before select() is called though you receive a SIGINT, your signal handler simply sets a global flag that is checked when select() returns (with errno == EINTR). So now the flag is set, the handler has done its job, and select() gets called and your program now starts waiting on file descriptors.
What you really wanted to happen is that the program would see the flag, clean up nicely and quit. Now the user has to send a second Ctrl + C to interrupt the select() syscall and to have the code executed that checks for errno == EINTR and the global flag that was set in the signal handler.
This is but a simple example of a race condition that can exist. The SIGALRM example given by the other HN user is also an excellent example of when things can go awry when not intentioned, and unless you program your signal handlers with that in mind you may get results you weren't expecting.