After encountering several pub quiz questions on the subject, I devised a mnemonic for remembering the order of the animals in the Chinese lunar calendar:
"Sheila, my Chinese doctor, presents rare oranges to rabbis driving skinny horses."
Broken down:
Sheila - Sheep (or Goat)
My - Monkey
Chinese - Chicken (or Rooster)
Doctor - Dog
Presents - Pig
Rare - Rat
Oranges - Ox
To - Tiger
Rabbis - Rabbit
Driving - Dragon
Skinny - Snake
Horses - Horse
Notice that several animals start with the same letter, but that there's enough matching letters in the mnemonic that it's not ambiguous - best to write the whole thing out though to make sure you have exactly one of each animal - this isn't Noah's Ark.
To relate it to an actual year, remember that 1919 was Sheep. (Since I made it up myself, I selfishly set it up so that Sheep is my birth year 1979, but 1919 works too and shouldn't be hard to remember). Don't forget, though, that the Chinese New Year is some time in late January or February - I was born on 17 January 1979 which was still the Chinese Year of the Horse, even though most of 1979 was the Year of the Sheep.
It's easier to remember if you can picture it - imagine Sheila, my Chinese doctor, in her white doctor's coat, kneeling in the road reverentially presenting the beautifully arranged plate of rare, delicious oranges to a bemused party of rabbis, who have long beards and serious expressions, at odds with their slightly ridiculous appearance, driving, as they are, emaciated skinny horses.
That's how I like to think of it, anyway.
Thanks to yclept for suggesting 'skinny horses' rather than 'shire horses' to avoid sheep-related confusion.