Carrie Mathison is an agent with the CIA.
Prior to going to work for the Agency, Mathison had been an Arabic language student at Princeton and it was there that she was approached and eventually recruited by a hardened agent named Saul Berenson. Upon graduation, she took the job with the CIA and began to earn a good reputation for her keen intellect and insight-fullness.
While she was attending university, she was diagnosed with bipolar disorder. She hid this situation from the Agency and manages to maintain a balance, sometime precariously, by quietly taking clozapine which she gets with the help of her older sister, Maggie.
As the series opens, Mathison is in her early 30's, making her career thus far having been probably just shy of a decade. The dates, according to the Homeland wiki, would indicate a speedy college transition as she was born in 1979 (in Maryland) and graduated from Princeton with a BA in Arabic Languages and Literature in 1979, having already earned a Bachelor's degree in Near East Studies and Linguistics from Northwestern and studying Arabic and Farsi in the American University of Beirut. However she accomplished all that, it clearly shows a very sharp mind and a determined attitude.
Working against that impressive brain, of course, is her bipolar problems which she medicates to maintain balance, sometimes having to add Lithium to the mix. She also, over the course of her time with the CIA, has had trouble with alcoholism, likely brought on as she fought the main problem.
Causing her some additional trouble is a rumored affair she had early in her time with the CIA with her boss, David Estes, which might have brought about his divorce. Certainly the current relationship is interesting at times.
Mathison is a dedicated intelligence officer who is willing to step outside the lines to get the job done, risking her career and even her life to get to the truth.