A field is a structure that behaves very much like the reals (under the usual operations). That is, it's a set of elements with an addition operation with an identity called 0 (special elements are named after their numeric equivalents, but that doesn't mean they're always the same thing), a multiplication with an identity called 1, and the two interact "properly". That is, they distribute and the additive identity is the multiplicative zero. Further, inverses exist for all elements, with respect to both addition and multiplication (except 0).

So very reals-like. Finite fields are then fields with a finite number of elements. Can you think of any examples?

The easiest way to read the OP is to read the field F as if it were the reals, except with a finite number of elements.

Let me know if that helps.