A professor agreed to write me a letter of recommendation for a military program. He said 'yeah, send me a vita'. What is a vita?
This is simply shorthand for curriculum vitae, also called a CV or resume. He's just asking you to send him your resume so he can list stuff you've done when writing your letter.
A "vita" is just a short term for a curriculum vitae (i.e., CV) which is similar to, but not exactly the same as, a resumé.
There are many questions on this site, like this one asking about how to create a CV and many other webpages which can help with figuring out how to write one.