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. 

Added in response to Kevin Lin's comment: In classical alg. geometry, people care about general fibers. The scheme theory provides generic fibers, which are really very convenient to have, since they are so concrete. The way "generic to general" usually works is as follows: You prove that the generic fiber has a property P, and that the property P is constructible. Then P holds for any $b$ in an open neighborhood of $\eta$, that is for a general $b$. EGAs contain a long list of properties which are constructible in proper (e.g. projective) families: smoothness, CM, normality, etc., etc. 

(And, yes, similar things were discussed in multiple other MO questions. One thing MO seriously lacks is a clear organization of the accumulated knowledge, so that people do not constantly ask and answer variations of the same question.)