Luna
Luna (which means "Moon" in Spanish) first appears in "A Moon Star is Born" (Series 1, Episode 1) when she realizes that Usagi/Serena is the one she has been looking for. (If you want a brief summary of their first meeting, check out Usagi's character profile.) Their relationship is not the best out of the Senshi group (Luna, in my opinion, gets along best with Ami/Sailor Mercury), but they do care a lot about each other. Luna clashes with Usagi most about the Senshi leader's daily habits (eating like a pig, not caring about school, being irresponsible, and *always* being late) but she usually means it for the best.
Luna's job is to be an advisor to the Sailor Senshi, but especially to Sailor Moon. Luna is the only cat (out of the three) that, to my knowledge, has any special powers. Luna can restore the Senshi's memory with the Luna Mind Meld and holds the senshi power/transformation pens as well as other gadgets they need (i.e. Mercury's computer) until the right time. Like all the Senshi cats, Luna can talk.
FYI: Luna turns human in the Sailor Moon S movie. To see a picture of all the cats in human form, check out the Cat Picture Gallery.
Artemis
Artemis (named after the Greek goddess of the moon and the hunt, why he has a girl's name I know not but SailorV once called him a transvestite) is Sailor V/Venus' kitty advisor, much as Luna is to Sailor Moon. He often gets on Minako's case about being clumsy and ditzy (Minako is clutzier and more like Usagi in the original version than she is in the NA dub) and makes fun of the girls more than Luna does. Like Luna, Artemis can talk.
Although it is totally cut out of the NA dub (can you believe this?), Artemis has a crush on Luna and has been known to make passes at her ;) This is significant because of our next furry feline friend...
Diana
Diana (named after the Roman goddess of the moon and the hunt and identifies with the Greek Artemis) is the future daughter of Luna and Artemis. I don't know much about her except that she accompanies Rini when she goes back to the past for the second time (to train to be a Senshi).