Most people think of the handle of a sword. In horseriding terms, however, the pommel of the saddle is the upper front of the saddle. Western people call this the "swell" or "fork." In a western saddle, the pommel or fork supports the horn.
In an English or saddle seat saddle, the pommel forms a loop over the horse's withers. It should not touch the horse's withers - in fact I have seen a situation like that cause a horse to rear, which can be very dangerous.
The pommel also supports a saddle bow, which I'll talk more about tomorrow.