I will delve a bit more into fomrations and such on offense first, as that is what Tengu initially asked about.
The formations I'll explain mainly come from the package that Red Night already described: (2 WR, 1 TE, 1 RB, 1 FB). One of the most common formations seen from this set is the I-formation. The QB lines up under center (under center means, directly behind the center), the fullback about 3 yards directly behind him, and the running back about 2 yards behind the fullback. This is a solid running formation; the usual procedure is, the QB receives the ball from center, hands it off to the running back, and the running back follows the fullback into the running lane that the offensive line is hopefully creating. This formation can also be used as a passing formation, typically in misdirection plays. The QB may fake a hand-off to the running back (called a play-action fake), which if done well, may freeze the secondary (the defenders who play the furthest downfield; they are the last line of defense), enabling a wide receiver to get open downfield.
Before I go any further, I want to explain what the tight end does. The tight end lines up, typically to the right side of the right tackle. He is an eligible receiver, and on run plays he typically blocks along with the rest of the offensive line. On pass plays, he goes out as a receiver. The tight end is bigger than a wide receiver and harder to tackle, but slower, and probably with not as good of hands. He is also obviously a better blocker than a wide receiver.
Now I will explain strong side vs. weak side. In sets that have only 1 tight end, the side that he lines up on is called the strong side, and the other side is the weak side. The reason is obvious; the side the tight end is on will have more blockers. Two variations of the I formation are the I-Formation Strong, and I-Formation Weak. In I-Formation Strong, the fullback is offset to the strong side, and vice versa for I-Formation Weak.
Another popular formation out of this set is Split Backs. In this formation, the running back and fullback are lined up side-by-side behind the quarterback about 2 or 3 yards apart. The fullback is typically on the strong side. (who can guess why?) This is an effective all-purpose formation, and if an offense has been successful at establishing the run (meaning, demonstrating that they have the capability to run the ball effectively), the defense genuinely won't know what type of play to be prepared for.
The last one I will discuss is the Shotgun. Normally the QB is directly under center, but in the Shotgun, he lines up about 3 yards behind center. This formation is generally used on 3rd and long (7 yards or greater), when there is no point in trying to make the defense think there may be a run play. Also the offense will typically sub out the fullback for a 3rd wide receiver. The advantage of the Shotgun is that it gives the QB a bit of a head start in dropping back to pass, and gives him an extra second or two to make his reads (determine which receivers are covered, which are open, and how the defense is reacting). The disadvantages: it makes it obvious to the defense that a pass play is on, and there's more of a chance of a bad snap between the center and QB.
The last thing I want to talk about is the concept of misdirection, the process of making the defense think one thing while the offense does something else. The aforementioned play-action fake is an example of misdirection, as is the draw play. In a draw play, the QB will take the ball from center, and begin to drop back as if to pass, while the running back acts like he is going to pass-protect (blocking to protect the QB from pressure while he's trying to pass the ball). Then, the QB will hand the ball to the running back. If the play works well, it will "draw" the pass rushers in, creating a good lane which the running back can burst through to gain some good yards. It typically will only work if the defense is expecting pass.
Anyway, I hope this explanation wasn't too dry. I tried to make sure I defined terms that would be unfamiliar to a lot of people, but if you have any questions about any of them, let me know. I will attempt to tackle defense a bit later, unless someone else does it first.