Weak alternator

Your alternator actually has little to do with starting the engine. If the battery has charge in it, then it should crank over. I would start with checking the battery cables on all ends and making sure that (1) there is no corrosion (usually a greenish white chalky looking gunk) at the connection points and (2) that said connection points are tight. If you can move the cables, then the connections are not tight enough. You might also replace the battery terminals if they have not already been replaced. If none of that solves the problem, then you will probably need to pull the starter and have it tested at the local auto parts place (should be free to test it). Let us know.

Also could just be a weak battery, but I kind of doubt it. Still take it to the auto parts store and test it to rule it out before taking the starter out.