Hey, and welcome
I usually say no to pre-built computers, but if you aren't up to making one yourself, lets check the specs:
Yeah that seems pretty good actually.
That video card is a replacement for the Geforce 6200 TurboCache that was released not long ago, which could achieve playable levels of gameplay for a very low price, and this ones specced at twice the speed.
So yes, it will play games, but not at high settings/resoloutions (Though it is Direct X9c compatable, with 4 pixel shaders, and 3 vertex shaders)
The reason these cards are alot cheaper is because as well as containing a little bit of their own memory, they borrow memory from your RAM, giving you even more.
So with that Mesh Computer, you could have the card take about 512MB from it to play games out of the 1024mb available.
So its actually a very good card for the price it costs, and will play some games at playable resoloutions with features new games make use of (Like High Definition Range (HDR) Lighting), though I don't know how fast this card would cope with that))
Its definietly also very good for the rest of the things you want to do, and is completly upgradable in the video card department to a better PCI-E chip when neccessary.