contingency-fee or contingent fee?

Or they are both acceptable.

Both are acceptable.
In the UK I believe they use conditional fee.

I often saw "contingency fee" on many legal web sites. And my Oxford dictionary also gives me the same variant

what is it anyways

?

a legal term meaning "an amount of money that is paid to a lawyer only if the person he or she is advising wins in court" (Oxford Dictionary)