A branded PSU is typically a PSU you would buy separately from a case and is of a reputable standing within the hardware community. Examples include Antec and Enermax.
A generic PSU is typically one included with a case (though sometimes good ones are included, but you have to be careful), or simply some really cheap one not from a good brand name.
The difference is that when a branded one is rated at 350W, it typically truly does output 350W. A generic one rated at 350W is not dependable and quite often outputs a lower amount of power, which is a huge problem. If you think your PSU is putting out 350W and it is really only doing 300W, that can be trouble. The PSU is not a good place to skimp on at all.
In this case, a branded 350W PSU should be fine from my experiences, though not by a whole lot. If you want a bit of breathing room, you should need no more than 400W, assuming you are not running a ton of extra hardware. The only reason to have more is if you are paranoid or cram lots of extra crap into your rig. I have a 450W PSU and that is way overboard already, and those people with 550W PSUs have really gone balls to the wall.