Is the prime P is fixed for an elliptic curve defined over a particular prime field F_p? I have seen the NIST, SEC and Brainpool standards. They have used same prime for a particular bit curve (128,192,256,521). Is the prime value fixed for a particular security (field size)?
In general you start by fixing the field, which translates into fixing the prime, and then you start to look to a suitable (i.e. secure/safe) elliptic curve defined over that field.

Note that, once you've fixed $p$, there are $\sim2p$ isomorphism class of elliptic curves defined over $\mathbb{F}_p$, which grants you a big set where you are supposed to find a curve with all the desired security properties, unless you are looking for very specific properties such as low CM discriminant.

There are two good reasons for it. 

First: the field is linked to the curve's cardinality by Hasse's Theorem. which means that by choosing the size of the prime you also strictly bound the size of the cardinality (within a bit). 

And second, and most important, a particular choice of the prime provides faster-than-generic modular reductions, which provides performance benefits of elliptic curve computations.

For example, all NIST P curves are defined over Solinas primes (also called generalized Mersenne), except P-521 which is over a Mersenne prime. 
The idea is that using Solinas reduction is faster than a generic modular reduction and can be used to speed up the computation.

Similar arguments are used in more modern elliptic curves such as curve25519 which uses a pseudo Mersenne prime ($2^{255}-19$) or FourQ which uses is defined over a field $\mathbb{F_{p^2}}$ with $p=2^{127}-1$.

Note that the Brainpool standard uses random primes, however its deterministic algorithm for curve generation selects first a random prime, and then look for a suitable curve. See appendix A of RFC 5639.