It does come in different gauges but it is hard to find much variety.  I don't know what your Home Depot stocks.  I saw 12, 19, and 23 gauge on their website.  What you want is the welded wire, not the woven or twisted.  A big raccoon or some other animals can pull holes in the woven or twisted where they at least have to break the welds to get through on the welded wire.  That is why you really don't want chicken wire.  It is twisted, not welded. It is also usually pretty fine gauge so some animals can break it as well as untwist it.  I think it was Tala that had some good pictures of both failure modes with chicken wire, from dogs.  
You can build something that nothing can get through, but that can cost you a fortune.  You can build something that will keep most things out at a more reasonable price.  From what I saw at the HD website, I'd personally go with the 19 gauge, but it is your choice.
I used 1/2" or 1" hardware cloth to cover my window and the vents, probably in the range of 19 gauge.  I really don't remember exactly what I used.  I used 2"x4" welded wire for my run and put chicken wire on the bottom 18" or so of the run.  That was more to keep baby chicks from going through the 2x4 wire, but also to keep the chickens from sticking their heads out where they could get taken off and to make it harder for something to reach in.  Mine sleep in a closed coop, not in the run, so I am not worried about reach-through higher up.