I know the definition of it, but is there a more intuitive way of thinking of it that even a layman could (kind of) understand?
The $n$-th homology group, for example, can be thought of as the number of $n$-dimensional holes in the space. The there something similar for the reduced homology groups?
The first thing to note is that the only homology group that will change is $H_0$. With that in mind, I think reduced homology actually better captures the intuition of "counts the holes in the space." Thinking that way, $H_0$ should count the 0-dimensional holes, which are just "gaps" between path-connected components. A point has no such "gaps," so its reduced $H_0$ is the trivial group. Fixing any particular base point more generally, the reduced $H_0$ counts how many different "gaps" there are between that point and the rest of the space (and adds a copy of $\mathbb{Z}$ for each one).