A round is a piece of music that loops back on itself. If you have a 12 bar round with an entrance point every four measures it would be sung in three voices. The first voice would start at the beginning, and when it finished the fourth measure the second voice would come in at the beginning, when the second voice was finishing the fourth measure the first voice would be finishing the eighth, and then the third voice would come in at the beginning. For this to sound nice the first measure has to harmonize well with the 5th and 9th, the 2nd with the 6th and 10th, etc.
An example of a trivial four bar round in three voices with a new entrance every measure/bar would be:
CLTF: =c8 e8 g8 /c8
Layered with all three voices it would become:
CLTF: =c8 [c]8e8 [ce]8g8 [eg]8/c8 [=g]8/c8 c8
You can see that this trivial round is just an arpeggiated Cmajor chord. And it's pretty boring.
It is useful to have a repeating chord progression to base your round on though. It makes creating the looping harmony much simpler. As an example let's try to write a simple round based on the progression Cmaj Fmaj Gmaj Cmaj. We'll make it six measures long with entrances every two measures. To fit in our progression then, we'll make each chord cover two beats of a 4/4 measure. So our round is going to be based on the chords:
Theoretically we can pull our first two beats from the first chord, the second two beats from the second chord, etc, repeat this process twice more and we'd have our six measure round. Some rounds created in this method will clearly sound better than others, but since when the round repeats we've guaranteed that the only notes sounding at the same time will be from a standard major chord we shouldn't get anything that sounds too bad.
Let's try that naive process then.
Here's what I came up with:
<1-2> e8A/  =g8EG
<3-4> [e]8/CE[=a]/  [=g]8b8[g]8EG
<5-6> [/c][=e]8E[/e]=eg[/f][=a]8A[/c]C  [=bg]8/D=B[/c=g]8EG
<7-8> [/c]=E[/e]=eg[a]8/FC  [=b]8/D=B[g]8/c8
<9-10> =Eega8  /D=B/c8
Actually I don't think this sounds too bad. I admit I followed some other rules of music composition, like repeating patterns of note durations, and ending on the tonic, (C, since we're in the key of Cmaj), although I intentionally didn't start on it. Maybe I should have just rolled dice.
:-) Then I probably would have gotten some odd skips in the melody, but there still wouldn't be any clashing notes since all the notes playing at any time are all members of a major chord.
You notice that I didn't use only half notes. You can break up the chords into whatever note values you want, as long as they add up to the time value you've assigned them in the chord progression, in this case two beats.
So to summarize what you need to do to create a round using this system:
Once you've got that basic frame work down there are some heuristics that I think make rounds sound better, and some deviations from the rules.
My first heuristic is occasionally pulling pairs of the voices into unison. When you're listening to the round, often your brain is listening to one voice at a time, and that lets you create an interesting crossover between voices. I also like the simplification and then expansion of the harmony.
Another thing I think sounds good in a round is if you can follow the voices separately. If they're all on top of each other and weaving back and fourth, that can create some nice harmony, but it can be hard to follow the round unless you're playing it on several different sounding instruments. If you let the melody follow broad tonal arcs and only merge and cross occasionally then it's easier to follow what's happening. I expect everyone likes a different mix of simplicity and complexity in their rounds, in any case, the weave texture is a good thing to keep in mind when you're writing.
Third, unless you want your round to sound like a hymn, you shouldn't have all of your notes transition at the same time. One voice could be singing a half note and then two quarter notes, while another is singing two quarter notes and then a half note. Like pulling the voices into unison sometimes, it's nice to have sections where all/some of the voices do follow the same pattern of note values, it can emphasize a theme and pull things together.
The important deviation to master is throwing in transitional notes that aren't in the chord you're pulling notes from. This is important to create a smoother/more interesting melody line. When you throw in transitional notes that aren't in the base chord you have to make sure that they won't clash with any of your repetitions though. How you do this depends on your work flow. I'm not a brilliantly foresightful composer who plans everything out from the beginning, so what I do is use Melody Assistant, which is a cheap composition program, and keep each voice of the round offset on its own score line. Then I can look at how things are overlapping, and make sure none of my transitional notes don't clash. Generally it's ok as long as you don't have two notes that are only a half or whole note apart striking at the same time. Of course there are other intervals that sound out of place too. But this isn't really a tutorial on composition.
If you're having trouble coming up with a chord progression, check out this nifty chord progression chart in the key of C by Steve Mugglin, as well as the rest of his Music Theory For Songwriters.
To send you off, here's one last round: (well, ok, it's not written yet. :-P)
If you're interested in the construction of rounds, you might also be interested in Timothy Smith's Anatomy of a Canon and his Anatomy of a Fugue