From http://jeff560.tripod.com/f.html:
FIBER, FIBER BUNDLE, and FIBER SPACE. According to J. Dieudonné A
History of Algebraic and Differential Topology 1900-1960 p. 387, the
terms “fiber” (German “Faser”) and “fiber space” (“gefaserter Raum”)
probably first appeared in Herbert Seifert “Topologie
dreidimensionaler gefaserter, Räume,” Acta Mathematica, 60, (1932),
147-238. However, Dieudonné adds that Seifert’s definitions “are
limited to a very special case and his point of view is rather
different from the modern concepts.” The modern concepts appear in the
1940s principally in the work of Hassler Whitney. Whitney defines a
fibre-bundle in “On the Theory of Sphere-Bundles”, Proceedings of the
National Academy of Sciences of the United States of America, 26, No.
2 (Feb. 15, 1940), p. 148. Within a few years the related terms
bundle, fibre and fibre space appeared: see N. Steenrod The Topology
of Fibre Bundles (1951). In recent years the spelling fiber has become
usual, conforming to common US usage.
Wikipedia adds that Seifert's definition didn't include the base space, and that Whitney first considered only spherical fibers.
Interestingly, the Hopf Fibration was named after Heinz Hopf. His original work, Über die Abbildungen der dreidimensionalen Sphäre
auf die Kugelfäche from 1931 describes a surjective map $S^3\to S^2$ that is not nullhomotopic. I cannot find a reference, but I would not be surprised if Whitney put it into the language of $S^1$ bundles.
In modern terminology, a fiber bundle is a map $p:E\to B$ with some additional properties, and each fiber is the inverse image of a point in the base space $B$. It is not too big of a leap to say that the fibers of an arbitrary map are the inverse images of points.
For groups, a map $p:G\to H$ gives a decomposition of $G$ into fibers, each of which is a coset of $\ker p$. If $G,H$ are topological groups, this indeed decomposes $G$ as a fiber space.
Another example is $p:SL(2,\mathbb{R})\to\mathbb{R}^2-\{(0,0)\}$ defined by $p(A)=Ae_1$. The fibers of this map are cosets of the stabilizer of the vector $e_1$, and one can check these fibers are 1-dimensional. This decomposes $SL(2,\mathbb{R})$ as a fiber space, and furthermore shows that the group is homeomorphic to a solid torus.