Yid is the Yiddish word for Jew.
Yid is pronounced 'yeed'. If you pronounce it to rhyme with 'did', it is offensive. To the best of my knowledge, such a pronunciation is offensive only because anti-semites (intentionally) mispronounce it that way.
Yid comes from the Hebrew name of the Biblical 'Yehudah' (in English, Judah), whose descendants formed one of the tribes of Israel. This name was later given to the Jewish Commonwealth at the time of the Second Temple. It was later shortened to the German 'Jude' (meaning Jew), and was taken into Yiddish in its current form.
Yid can refer to either a male or a female, and is a neutral term. The feminine of yid, yideneh, is always used as an insult or rebuke.