Musical form is nothing new. Many modern composers, like Eric Whitacre, tend to write pieces that don't follow a set form, but that's because modern composers are pretty weird that way.
You're all already familiar with the concept. Popular music has verses, bridges, choruses, an introduction, etc. You all instinctively know when the chorus is coming up in a song, just because you're all so familiar with pop music form. Poetry, too, has form:
My candle burns at both ends,
It will not last the night.
But ah, my foes, and oh, my friends,
It gives a lovely light!
~Edna St. Vincent Milay
If you look at the poem, you'd say its rhyme scheme was ABAB, since "ends" rhymes with "friends" (lines 1 and 3, respectively) and "night" rhymes with "light" (lines 2 and 4, respectively). Likewise, when I use A and B when talking about musical form, it's the same idea.
The first form I'd like to show you is alternatively called Sonata Allegro form, Sonata form, or First Movement form, because it's the form most commonly used for the first movement of symphonies. Here's basically what Sonata Allegro form looks like:
Introduction - A - B - C - A - B - C - !!!OMG!!! - A - B - C
Don't worry, that will makes sense in just a little bit; I've got a recording here to show you what I mean.
Here's the first movement from Beethoven's Fifth Symphony. It's the freaky butterfly one from Fantasia 2000, although I guarantee you'll all recognize it from the first four notes:
A-B-C, A-B-C, !!!OMG!!!, A-B-C
0:06 - 0:13 -- Told you you knew this piece! This is the introduction.
0:13 - 0:28 -- This is A, the first theme, and the start of what's called the Exposition. Like you're being "exposed" to the themes.
0:28 - 0:49 -- This is, essentially, the " - " between A and B.
0:49 - 0:59 -- Here's B. B is usually very peaceful, quiet, delicate, and happy sounding.
0:59 - 1:12 -- And here's the " - " between B and C! Isn't it cool how B just blended right into " - "?
1:12 - 1:20 -- Here's C! Usually, C is the end of the Exposition (the Exposition is "A - B - C"), but Beethoven decides, for good measure, to do just a little bit more before he finishes the exposition:
1:20 - 1:28 -- Beethoven decides to wrap up the Exposition by tossing in some stuff that sounds similar to A. Now we're officially done with the Exposition! Know what that means?
It means we're going to do it all over again!
1:29 - 1:36 -- Sounds exactly like the Introduction at 0:06, doesn't it?
1:37 - 1:53 -- Here's A!
1:53 - 2:14 -- Here's the bridge to B! Sounds familiar now, hmm?
2:14 - 2:24 -- And here's B! Right up until the flute finishes his nice little solo.
2:24 - 2:37 -- Bridge to C.
2:37 - 2:46 -- Here's C!
2:46 - 2:53 -- And here's that cute little ending Beethoven added in, which means we've finished the Exposition for a second time!
Now comes the !!!OMG!!! part. This is what's called the Development. In the Development, the composer takes fragments and pieces out of the Exposition and just plays around with them. It also has a tendency to be very dark, and to feel like you're lost in the woods or something. It can also be kinda intense, edgy, or dissonant. The Development is (to give the textbook answer) "characterized by tonal instability and fragmentation".
It'll be like you're lost in the woods, hearing little bits and pieces of the Exposition being thrown at you.
2:55 - 2:59 -- See? It sounds like the Introduction... but kinda... different...
3:00 - 3:08 -- ... sounds just like A. I thought you said the Development would be different?
3:08 - 3:31 -- Right at 3:08 you should be like, What's going on? Where are we going? Am I gonna hear B? WHERE AM I?! WHAT'S HAPPENING?!
3:31 -- Is it... B...? It sort of is...
4:04 -- That's... kinda like the beginning of B and the Introduction... What's happening?
4:17 - 4:25 -- OMG! It's the Introduction again! It's so huge and triumphant, like we've just left the forest and found civilization again!
This is what's called the Recapitulation. Basically, it's the Exposition one last time... but with a few differences.
4:25 - 4:36 -- Here's A again, just like before!
4:36 - 4:50 -- Wait... oboe solo? That wasn't in the Exposition, was it? The biggest difference between the Exposition and the Recapitulation is that the " - "s are going to be very different.
4:50 - 5:11 -- Well, here's the old " - " between A and B.
5:11 - 5:23 -- Whee! It's B, and it's virtually unchanged, except now the violins are playing it instead of the clarinet and flute.
5:23 - 5:37 -- And here's the " - " between B and C, just like we remember it!
5:37 - 5:46 -- It's C!
5:46 - 5:54-- And here's the little ending that Beethoven put at the end of the Exposition. Piece is over, time to clap and go home!
5:54 -- ... wait... he's still going? Wasn't it supposed to end right there...?
"Screw you," Beethoven is saying, musically, "I'm not done yet!"
Now, if you're interested in knowing what my absolute favorite part of this entire piece is, it's right here, from 6:09 - 6:33, specifically.
So there you go. Exposition - Exposition - Development - Recapitulation. Or, more simply, ABC - ABC - OMG - ABC.