what does it mean that something fibers? For example, in an article I have found that "compact abelian group which fibers over the circle $S^1$ [...]" and  surely I have heard that phrase before.
What does it mean?

There are various kinds of maps $f: E \to B$ called fiber bundles.  They occur in topology and algebraic geometry (I'm guessing on that last one).  Saying that $X$ fibers over $B$ usually means that there exists a bundle $f:X \to B$.  
Topologically, a fiber bundle $f: E \to B$ with fiber $F$ is a continuous map of spaces with the following long-winded property:  to each point $b \in B$ there is an open set $U$ about $b$ such that $f^{-1} (U) \cong U \times F$ canonically through projections.   Some people might impose additional conditions on connectedness, smoothness, the structure of $F$, etc.  This all depends on the needs and goals.  For example, you may want everything to be a topological group, which sounds like your situation.
The standard first examples are (a) product projections $p: B \times F \to B$ and  (b) the exponential cover $p: \mathbb{R} \to S^1$ by $p(t) = e^{2\pi i t}$.
