In most cases it is necessary for the fixed point F to have known prime order. A little bit of algebra shows why this works. The finite field needs to be chosen so that the prime order of the fixed point is sufficiently large, say 150-300 bits. A field representation defines what bit-patterns are used to represent the various field elements. An analogy is using different representations for integers - for example base 10 or base 2.
Polynomial basis is probably easier to understand.
With a polynomial basis, field elements are represented as polynomials. This is analagous to performing multi-precision arithmetic on words instead of bits. It is always possible to find a field element, called a generator, such that the whole finite field is generated by the powers of the generator. For polynomial basis, one looks for trinomial reduction polynomials to improve the speed of modular reduction. For normal basis, one looks for an 'optimal' normal basis such that field multiplication is efficient.