Best is a matter of preference, you should specify more criteria for what "Best" really means for your build.
Within that price range I would recommend you consider these four things if you are building a budget *gaming* PC:
- Memory Interface (a.k.a. Bus width)
(between USD$100-150, 256-bit is the best available)
- Memory Type
(you will find either GDDR3 or the better GDDR5 in that range)
- Memory Size
(512MB to 1GB is the most you'll get with 256-bit Memory Interface in that price range)
- DirectX Generation
(You will have a much easier time finding higher quality DirectX 10 cards in that price range than newer cards that support DirectX 11)
That all said, for most gaming and video applications, I would recommend a Radeon HD 4870 card, because I have used one myself for over a year in my desktop. The one I have is a 256-bit Memory Interface, 1GB GDDR5 DirectX 10.1 card that has provided me a lot of high definition joy in many 3D games. The model I have is no longer available, but it was made by XFX.
There are two equivalent Radeon HD 4870 cards available at Newegg made by ASUS or HIS within your price range, the ASUS currently cheapest at $130 before rebate with free shipping in the US.
I make that recommendation if your choice is going to be a single video card (which, if you ordered a budget motherboard with only one PCI-E 16x 2.0 slot, is your only choice).
Joe88 linked a Sapphire Radeon HD 5770, which is a newer generation card that supports DirectX 11. It also has 1GB GDDR5 which is excellent, but its Memory Interface is 128-bit. I think this is an excellent alternative if you think you might soon buy a second identical card to run a Crossfire setup, or if you genuinely prefer DirectX 11 support to memory performance. Note this card may also require more power.
The difference between 128-bit and 256-bit Memory Interface is a big one, but that is also a matter of the preference of the person you're building the system for. If you're going to be running a game that doesn't really use DirectX 11 (such as World of Warcraft) you will be able to get higher framerates at max settings with the 256-bit card. If you're running a much newer 3D game, more processing will be necessary and the bus width is no longer the only bottleneck (that's where having more processing power may give you competitive or better performance).
