How wide of tires can I run on a Victory Vision road bicycle? I'm new to road bicycles, sorry if this question is beat to death. What I'm wondering is what to measure as far as frame clearance, rim width, etc. to put the 'widest tires that will fit' on my road bike for durability purposes? 

The bike is more or less a glorified BSO, and because of this I can't find any real specifications other than the amazon.com page: http://www.amazon.com/Victory-Vision-Mens-Road-Bike/dp/B002D126EW



The ones that came on it are unmarked, but appear to be 27mm or 28mm, the rim width is 25mm. How do I know whether 32mm or larger will fit before buying them, without rubbing and such?
What tire width can be safely used it mostly determined by the width between the bead seats in the rim.  This can be measured and is often listed in the specifications for your rims.  

Then I use Sheldon's Tire sizing chart to figure out which tires will work.  That link is to a long article on tire sizes, the rim vs tire width chart is red &amp; green and is near the bottom of the article

Edit

Near the top of the linked article is a nice picture showing where to measure inner rim width.  This determines if the tire will stay put on the rim.  Both too wide a tire on too narrow of a rim and too narrow of a tire on too wide of a rime will not seat correctly.

As freiheit points out you also need to check for clearance issues around chainstays, seatstays brakes etc.