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Budget GPU: AMD Radeon HD 6670 DDR3
A budget price gets you previous-generation technology, but the AMD Radeon HD 6670 is a very capable card which has the power to handle any modern game you throw at it -- just keep your expectations realistic and don't plan on cranking the settings all to max.
Remember to look for the DDR3 models of these cards, as you'll end up paying the extra for the faster GDDR5 RAM.
Price: From $59.
Image source: SapphireTech.