The basic act of casting a spell is to point your wand in the desired direction and say an incantation. As we learn in the famous "Wingardium Leviosa" Charms lesson, the way in which you point your wand (swish and flick!) and the manner in which you pronounce the spell can achieve different results. As a wizard or witch becomes more prolific, they get to the point where they are able to perform magic without saying the spell out loud (though, we see that Harry thinks the word for his desired spell). Another thing to note is that all of the spells in the series are based in Latin, and they are often simple calls to what the spell does. (i.e. lumos means light, and when used as a spell, it creates light). If you're the artsy type, Pottermore has a section of profiles for common spells, with beautiful illustrations and quotes from the book.