I am clicking on an object but it is not being selected...

There are several reasons that an object might not be selected if you click on it.  It could be a locked object (in which case clicking on it would cause nothing to be selected), a glued object (in which case the object to which it was glued and this object would be selected), or there could be something transparent between your view and it which is being selected in its place.

If, however, the object IS being selected but so are other objects then see this 

If an object is locked, click here for how to select/manipulate locked objects

If an object is glued, then to select just the one object you want, you need to first use the Unglue Objects function to separate them and then select the one you want using Shift-Click or any of the other selection objects described in the solution above.