1. To recognize someone or something, or to figure out who/what something is through other means.
Examples: The culprit was identified by a witness.
The body was badly mangled, but forensics was able to identify it using DNA evidence.
She identified the smell as being from curry powder, something her mother used in cooking when she grew up.
2. To relate to someone or something. Usually this is used as "identify with" or "identify as"
"Identify with" carries the connotation of "relate to" and "identify as" carries the connotation of "I have assumed this identity by choice."
Ex: Her experience as an only child resonated with me. As an only child myself, I identified with her story.
I identify with -insert fictional character here- because we have a lot in common and I see myself in them.
She identifies as a democrat.
He identifies as a Christian.
You may hear people say things like "I identify as gay/bisexual/etc" but I personally think this is incorrect. You do not "identify as" a sexual orientation. A person's sexuality is innate and intrinsic to their being, not something you can relate to on the basis of having similar experiences. You can identify with experiencing homophobia for example, but you cannot "identify as" being gay unless you actually are gay. It would be better in that instance to just say "I'm gay" instead of "I identify as gay." Being a democrat or a Christian are choices, which is why it makes sense to say you "identify as" those things. Sexuality on the other hand is not a choice, and I personally feel it's inappropriate to use "identify" to describe one's sexuality.