The algebraic closure of Fp has one really important property, which is that its Galois group over Fp is
isomorphic to the profinite completion of Z and is topologically generated by the Frobenius map x -> x^p. The fixed field of the subgroup nZ is precisely Fp^n; in other words, there is exactly one copy of each finite field of characteristic p and they are nested according to divisibility. Any statement you can make about a finite number of elements in this field takes place, as others have mentioned, in a sufficiently large finite subfield.
One interesting way to get a grip on the elements of the algebraic closure is to think of them as aperiodic necklaces. As explained in this question I asked, there is a (non-canonical) bijection between irreducible polynomials of degree n over Fp and Lyndon words of length n on an alphabet with p letters. One should think of conjugating as analogous to cyclic rotation, and the choice of a particular representative of a necklace up to rotation as the choice of a particular representative of a point up to conjugation. This is the categorification of the fact that as dynamical systems, words on an alphabet of length p and the algebraic closure of Fp have the same zeta function.
(The upshot of all of this is that some people think there should be a proof of the rationality of the zeta function of a variety over Fp using formal language theory, by finding a regular language that describes those points. Unfortunately, all known translations give non-regular languages.)