Celia is the daughter of Duke Frederick, cousin to Rosalind, and a general balance to Rosalind's foolish love.
Celia values her relationship with Rosalind so much that very little else matters to her, which is why she runs away with Rosalind to the Forest of Arden (disguised as Aliena). This makes sense, as the girls initially seem like twin sisters:
[...] if she be a traitor,
Why so am I. We still have slept together,
Rose at an instant, learned, played, eat together,
And, wheresoever we went, like Juno's swans
Still we went coupled and inseparable. (1.3.75-79)
Initially, Celia and Rosalind are partners in crime, jesting together, running off together, changing their identities together (Celia becomes Aliena), and generally having fun… together. Celia even declares their trip into the forest isn't banishment, but a chance at liberty, as they get to be the women they want to be (together).
Celia and Rosalind's relationship changes once Orlando enters the scene. Celia is relegated to dealing with Rosalind's constant lovesickness and hijinks as Rosalind/Ganymede. As the play develops, Celia is confined more and more to simply reacting to Rosalind's antics.
Through Celia, we get to investigate Rosalind's changing attitude toward love, which is contrasted by Celia's unchanging skepticism. Rosalind and Celia both see love as foolish fun at the beginning of the play, but things change when the girls get to the forest and learn that Orlando is there. It turns out that even though Rosalind thinks love is foolish, she's not above being foolish herself.
Celia, who is not in love, has to watch her friend embrace all the silly stuff they've spent their time mocking. In that case, it is no surprise that, throughout the play, Celia becomes more and more surly, perhaps disenchanted with her friend's enchantment. When Celia begins to tell Rosalind that Orlando doesn't really love her, the clear divide between the cousins becomes apparent. Celia perhaps resents Rosalind's attention to Orlando, but it seems she particularly resents that her friend takes the foolishness of love seriously.
Celia's disdain of love comes into perspective when Celia herself falls for a guy. Celia's relationship with Oliver isn't a particularly deep one, but it's significant in the play because it sets off how unusual Rosalind's approach to love actually is. For all Celia's pooh-poohing about love, once she finds a man we literally never hear from her again. She drops out of the play, totally enamored of her new boy. Celia then becomes a hypocrite of sorts; she loses herself in another person, and thus is lost to the world.
Rosalind may be giddy about being in love, but she's actually done a remarkable job of keeping her own identity in the process. Because she knows love is foolish, Rosalind can be in love without being a fool for love. Celia's disappearance highlights exactly what Rosalind didn't do, making Rosalind even more of an extraordinary woman. Unfortunately, Duke Frederick ends up being right—Rosalind one-ups Celia, and we all think she's a better catch than his daughter.