In other words, using the idea of reduced homology, It is common to consider such a space as a nonempty space without "holes"; for example, a circle or a sphere is not acyclic but a disc or a ball is acyclic.
This condition however is weaker than asking that every closed loop in the space would bound a disc in the space, all we ask is that any closed loop—and higher dimensional analogue thereof—would bound something like a "two-dimensional surface."
The condition of acyclicity on a space X implies, for example, for nice spaces—say, simplicial complexes—that any continuous map of X to the circle or to the higher spheres is null-homotopic.
If a space X is contractible, then it is also acyclic, by the homotopy invariance of homology.
Nevertheless, if X is an acyclic CW complex, and if the fundamental group of X is trivial, then X is a contractible space, as follows from the Whitehead theorem and the Hurewicz theorem.
For instance, if one removes a single point from a manifold M which is a homology sphere, one gets such a space.
For example, the punctured Poincaré homology sphere is an acyclic, 3-dimensional manifold which is not contractible.
With every perfect group G one can associate a (canonical, terminal) acyclic space, whose fundamental group is a central extension of the given group G. The homotopy groups of these associated acyclic spaces are closely related to Quillen's plus construction on the classifying space BG.
The converse is not true: the binary icosahedral group is superperfect (hence perfect) but not acyclic.