A modeling amp is an amplifier with solid state circuitry and possibly something called a DSP (digital signal processor) that simulates the sound you get from one or more different amps. I wrote a review of a small one I used to have some years ago here:
http://www.gad.net/Blog/2010/07/20/vox-vt30-valvetronix-amp-review/ Tech has advanced considerably since then.
The one Bluespicky linked is an amp simulator pedal which will require amplification. That may sound odd, but they're designed for running direct to the PA or FOH system. You'd probably want something self contained for simplicity's sake. That said, SansAmp kicks ***. They have some very cool options.
Brands: The most common you're likely to see is Line6. I'm not a fan for various reasons.
One that's getting a lot of attention lately is the Mashall Code amps. They sound amazing for not a lot of cash and the larger ones will sit with any band.
Vox still makes great modeling amps.
The Fender Mustangs get decent reviews, but I've never been thrilled with them.
There are a bunch more. As a guy who's played through most of the great vintage tube amps, I currently use an Axe-FX modeler which is orders of magnitude beyond any of the ones listed, and that includes the price.
For me, versatility and light weight trumps tone because I can get 90% of the tone and simulate 100 different rigs without breaking my back.
My advice is to go to a guitar center and try them all. If you don't feel comfortable playing, ask one of the GC sales guys to demo them. I'm sure they'd love to hear their own playing instead of Stairway to Heaven for the 384th time that day.