I will not presume that you asked about what a good cover is. Thinking about orbifolds the way you described is useful: a group $G$ acts on $M$ properly and discontinuously but because the action is not free the topological quotient $X = G\backslash M$ may be missing some information. In particular the map $M \twoheadrightarrow G\backslash M=X$ is not a covering map.

One definition of an orbifold is the space $X$ equipped with an atlas $\mathcal A$ such that each chart contains some extra information (i.e. a local group action.) This is the extra information that is lost, for example, in passing from $M$ to $X=G\backslash M$. With this information, you can recover $M$ from $X$ and $G$, should $M$ actually exist (this has to do with "goodness"). So in this sense an orbifold is a quotient of a proper discontinuous group action.

A covering space of $Y$ can be obtained by taking a open cover $\mathcal{U}=\{U_i\}$ of $X$ and glueing together copies of the $U_i$, one of the criteria for being a covering space though is that two distinct lifts of some $U_i \in \mathcal U$ are disjoint.

If you want to make an *orbifold* cover, i.e. get $M$ from $G\backslash M$, the problem is that distinct lifts of the same chart in $\mathcal A$ can *not* be made to be disjoint.

Suppose first that the non-trivial fixed sets of the $G$ action on $M$ are points (and that $M$ is a surface). Let $X = G\backslash M$ and let $\mathcal A$ be the orbifold atlas on $X$. Then $M$ can be obtained by taking copies of the charts $V_i$ in $\mathcal A$ (which are all discs), but *sometimes* you will need to slice some of the $V_i$ open (to get something that looks like a slice of watermelon), you can then get $M$ by glueing together possibly sliced charts from $\mathcal A$ in a manner analogous to building a covering space. Moreover, these "slices" in charts are exactly the cuts in branched covers. So in this case the orbifold cover is a branched cover.

However, as Andy Putman commented, not all actions of $G$ on $M$ have fixed sets that are points. For example a reflection will fix an arc, and the example that Andy gave ($\mathbb R^2$ modulo reflection) will not correspond to a branched cover. So to answer your question, branched covers are a way to construct special (but important) cases of orbifold covers. The special case, as Andy commented, is when all the non trivial $G$-fixed sets in $M$ are points (provided $M$ is a surface).