1) Playcalling. During this phase, each team's defensive or offensive coordinator chooses a play and signals his choice to a key player on his team. Each team then gathers briefly while the player passes on the word about what play's been called -- this gathering is called a huddle -- and, once the play has been called, the team breaks the huddle and takes up the appropriate positions.
2) Pre-snap reads. Once both teams are in formation for the coming play, the quarterback looks at the defense to try to figure out what it will most likely be doing, and how that will affect the play called by the offense. This is called "reading the defense," and, because it's done before the ball is snapped (more on that in a second), we say that the quarterback is making a pre-snap read.
Notice, though, that I said there are pre-snap /reads/, plural, going on. This is because, depending on the offensive system, several offensive players may also be assessing the defense and determining changes that need to be made to their plans. Moreover, at least one defensive player is charged with reading the OFFENSIVE formation and figuring out what, if any, adjustments the DEFENSE need to make.
Finally, if either the quarterback or the key defensive player determines that the play called in the huddle is completely borked, he can change to a new play by calling an audible -- a pre-arranged verbal signal that tells his teammates to run a specific different play.
But all too soon, it's time to run the actual play itself, which brings us to...
Phase 3: the snap. One of the offensive linemen, called the center, is always lined up directly over the football. The quarterback, moreover, is lined up directly behind the center; it's the center's job to get the football off the ground and into the quarterback's hands.
So, when it's time for the play to start, the quarterback signals to the center, and the center either hands the ball to the quarterback (if the quarterback is only a foot or two behind him) or throws it back to him (if the quarterback is several yards deep). In either case, this requires the center to make a very abrupt, violent movement of his hand to get the ball from Point A to Point B -- a sort of snapping motion, which is why, when the center puts the ball in play, it's called the snap.
Aaaaand now the play itself has, at long, LONG last, started.