Difference between homology and integral homology? What is integral homology? And how does it relate to homology? I can't find a good answer anywhere, so I thought I would ask here.
 A: Homology involves chains, boundaries, and cycles, right? And a typical cycle looks like
$$
c_1 s_1 + c_2 s_2 + \ldots + c_n s_n
$$
where the coefficients $c_i$ are in some group, and the $s_i$ are simplices of some sort. 
If the group used is the integers, you get integral homology; if it's something else (e.g., $\mathbb R$) you get a different homology. 
Finally, there are other things you can compute, like "all closed forms mod all exact forms on a smooth manifold", and the result is called "deRham cohomology", and is closely related to the homology groups of the manifold, with real coefficients. 
One feature of integral homology: it may contain torsion, i.e., groups like $\mathbb Z / 2\mathbb Z$, while homology groups with field-coefficients are all vector spaces over the coefficient field. This integer homology tends to be a finer invariant. 
A: You can take homology $H_{\bullet}(X, A)$ with coefficients in any abelian group $A$. Integral homology is the special case where $A = \mathbb{Z}$. "Homology" with no qualifier usually refers to this.
