Characterizing spaces with no nontrivial covers I know that simply connected locally path-connected spaces have no nontrivial covers. Is there a characterization of spaces with this property?

The converse is also true, under mild assumptions on $X$. 
Namely, if you assume that your space $X$ is path connected, locally path connected, and semilocally simply connected, then $X$ has no nontrivial path connected covers if and only if $X$ is simply connected. 
This follows from the standard theorem giving a bijection between subgroups of $\pi_1(X,p)$ and connected pointed covering spaces of $(X,p)$. Under this bijection, the degree of the covering map equals the index of the corresponding subgroup. So the covering map is nontrivial if and only if the subgroup is proper, and every nontrivial group has a nonproper subgroup.  
