Well, there is Iskovskih’s classification which is very explicit. See V.A. Iskovskih, Yu. G. Prokhorov: Algebraic Geometry V: Fano varieties. Encyclopaedia of Math. Sciences 47, Springer-Verlag, Berlin 1999.

Yes, Fano threefolds have been completely classified and one has explicit projective models of them. See V.A. Iskovskih, Yu. G. Prokhorov: Algebraic Geometry V: Fano varieties. Encyclopaedia of Math. Sciences 47, Springer-Verlag, Berlin 1999. See also Andreas Ott's thesis on Fano threefolds of picard number $\rho\ge 2$.

In some sense the most complicated Fanos are the ones with Picard number one, i.e., $Pic(X)\simeq \mathbb Z$. Here there are 18 families of such threefolds and they have been classified by Iskovskih. The most basic invariant here is the $index$ of $X$, i.e., the maximal integer $r$ such that $K_X$ is divisible by $r$ in $Pic(X)$. Fano threefolds of higher Picard number was classified by Mori and Mukai. This was one of the early triumphs of Mori theory.