An irreducible scheme $B$ has a unique generic point $\eta$. The generic fiber of a family $X\to B$ is the fiber $X_{\eta}$ over that special point $\eta$.
A general fiber $X_b$ is a fiber over $b\in B$ that belongs to some fixed open set $U\subset B$. And very general means that $b$ belongs to $V$ which is a complement of countably many Zariski closed proper subsets $Z_i$ of $B$.
That is the most common modern terminology. In older (and not so old) books sometimes generic is used where general would be more appropriate.