Is it "vita" or "vitae"? Why?

Vita means life. Vitae is the genitive (possessive) form, of life. Curriculum means something like course. So Curriculum vitae means the course of one's life, and makes good sense. Vita by itself also makes good sense, though it's perhaps less accurate as a description of the document in question. Neither curriculum vita nor vitae without curriculum makes sense.

Addition, suggested by Patricia Shanahan: All the words in bold italic are Latin.

"Vita" is an American English term, synonymous with "Curriculum vitae".

If your web site targets only Americans, then "Vita" is fine as a label. If you wish to appeal to an international audience, you'd be better to call it a "CV".

The phrase "Curriculum vita" is an error, and makes no sense. That's what the speaker at the workshop was telling you.

About the pluralisation - my instinct would have been to give "vitas" as the plural of "vita" (on the grounds that if it doesn't mean "life", it's not Latin, and therefore shouldn't get a Latin-sounding plural). However, Merriam-Webster gives the plural as "vitae". Since Merriam-Webster is a greater authority on American English than I am, I defer to its judgement. I am not a native speaker of American English.

The origin of this term appears to be the Latin "vita", meaning "life". Just to clarify the variations on this term in Latin ...

life           = vita
life's         = vitae
life's course  = curriculum vitae
life's courses = curricula vitae
lives          = vitae
lives'         = vitarum
lives' courses = curricula vitarum

If I have multiple CVs on my computer, which I use when I apply for different types of job, these are "curricula vitae", because only one life is involved.

If I have multiple CVs on my desk, because I am about to interview several applicants for a job, these are "curricula vitarum", because they are the courses of several lives.

Update

In response to someone asking me to support my claim that "vita" is an American English term.

"Vita" is found in Merriam-Webster, which is the canonical dictionary of American English. It is not found in the Oxford English Dictionary, which is the canonical dictionary of Commonwealth English (the edition I checked was published in 1971).

Most tellingly though, it is found in Oxford's online dictionary, where it is labelled as US.

The others have already explained the latin, but if you're still looking for what short word to put in your menu instead of "Vitae", I'd like to suggest just 'CV'.