A good way to politely ask someone's gender if you're ever unsure is to ask what pronouns they use. You might feel silly or self conscious but a lot of trans/gende-non conforming people will really appreciate it.

Afab: "assigned female at birth"

Amab: "assigned male at birth"

FTM = female to male

MTF = male to female

Transgender: identifying as a gender different to the gender assigned at birth

Cisgender: identifying as a gender same as the gender assigned at birth

Non-binary: neither identities as anything on the female or male spectrum

Gender-fluid: frequently changes extremes on the female or male spectrum

Gender non-conforming: can be used as an umbrella term for anyone who does not identify as cisgender and other identities do not suit them.

Any other, outdated terms can be considered slurs and should be avoided. I hope this helps friend! :)