"In that case," said the Dodo solemnly, rising to its feet,
"I move the the meeting adjourn, for the immediate adoption
of more energetic remedies-"
"Speak English!" said the Eaglet. "I don't know the meaning
of half those long words, and, what's more, I don't believe
you do either!" And the Eaglet bent down it's head to hide a smile:
some of the other birds tittered audibly.
"What I was going to say," said the Dodo in an offended
tone, "was, that the best thing to get us dry would be a Caucus-
race."
"What is a Caucus-race?" said Alice; not that she much
wanted to know, but the Dodo had paused as if it thought that
somebody ought to speak, and no one else seemed inclined to
say anything.
"Why." said the Dodo, "the best way to explain it is to do it."
Lewis Carroll - Alice's Adventures in Wonderland
Capter III - A Caucus-Race and a Long Tale
As with everything else in his rambling story Dodgson was rewording his tales to the Liddell girls - in the original story a bunch of animals climb out of a river and head to a house where they dry themselves around a fire. This was directly inspired by the events of 17 June 1862, when Dodgson, his sisters and aunt went on a picnic with Reverend Duckworth and the three Liddell sisters (Lorina, Alice and Edith). They ended up walking three miles in a sodden downpour to a house owned by friends of Dodgson. In the original version of the story Dodgson has the Dodo (himself) take the Lory, Eaglet and Alice ahead to the house. The Duck is the one telling the Dodo not to use big words.
These animals are of course Dodgson, Lorina, Edith and Duckworth - the Mouse is apparently the girls' governess, Miss Prickett (the history lesson is lifted from the girls' history book). In changing the scene Dodgson was merely trying to remove the more personal version of the story he wrote for the Liddell girls.
What we end up with is the Caucus-race, which seems to have been added so adults (children probably too, at the time) can laugh at a joke It is a joke, and a particularly good one. If we zip over to dictionary.com we can look up "Caucus" and find a bunch of entries, one of which says the following:
Chiefly British. A committee within a political party charged with determining policy.
Well that's all good, but why did he use the word? Apparently Martin Gardener, editor of The Annotated Alice suggested that he used this because committees spend a lot of time going in circles and getting nothing done - yet expecting a reward at the end. Though it's possible he used it as a nonsense word I think the Gardener explanation makes sense and it sounds like the sort of thing Dodgson would do - this is someone who works at a university and deals with bureaucracy.
As for everyone getting a prize, though it's been said that it might be Dodgson thinking of the Aeneid it's more interesting to compare it to today, when people get a small prize for participating in a team building exercise (not to mention our reluctance to only give one child something after a competition at school). What are the animals given? comfits - Victorian candy. It's like life imitating art - Wonderful.
Sources
An interesting note is how the Disney film portrays the Dodo as using a fire to dry himself while the rest run around in circles. I think they should have been told they were portraying real people....