The importance of memory for a graphics card is two fold. It's important to have enough memory to store all the data required to render a given 3D scene on board the graphics card itself. The alternative is dipping into the PC's main memory, and that means latency, lag and stuttering frame rates. Treat 512MB as a minimum for decent performance in modern games.
The other half of the story is the related issue of bandwidth. Keeping all those shaders and ROPs fed with pixel data takes some serious throughput. The latest cards therefore pack ultra fast memory chips that run as fast as 1GHz or more and are able to transmit data at least twice per cycle (hence the term DDR or double data rate). The latest GDDR5 (the G stands for graphics) is actually capable of four transmits per cycle. Bus width is another factor that affects bandwidth, the more bits the bus supports, the more data it can pump per cycle.
The biggest current bus is the GeForce GTX 280's 512-bit beast. However, large memory buses take up a huge amount of space on a graphics chip, so the introduction of GDDR5 memory will probably see bus technology scale back to 256-bit with the next generation of big GPUs.
The joker in the GPU pack is undoubtedly multi-GPU technology. Both of the big boys of PC graphics, AMD and Nvidia, offer multi-GPU platforms in the shape of Crossfire and SLI respectively. The idea is simple enough - to use multiple GPUs in parallel to provide even more rendering oomph.
When they work, the results can be spectacular. The problem is, all too often they don't and you are left with the performance of a single card or worse. Also note that special supporting motherboards are required and, in the case of Nvidia, that exclusively takes the form of an Nvidia motherboard chipset.
A final word, in terms of 3D performance, should go to integrated graphics as found on motherboards. In theory they offer the same feature set as discrete GPUs. However, in order to make integrated GPUs small enough and cheap enough for motherboards, the number of functional units is brutally cut down, typically by a factor of 20 or worse, compared with the fastest stand-alone solutions.
2D features: 2D acceleration
First up is hardware video acceleration. Here, the two big players are once again fairly level pegging. All the latest DX10 boards from both AMD and Nvidia have built-in 2D engines dedicated to accelerating modern and demanding codecs such as H.264 and VC-1.
2D features: video ports
VGA may have been revolutionary in 1997, but it looks pretty laughable compared to modern digital interfaces. Today, DVI remains the dominant standard on the PC and in dual-link form is good for up to 2,560 x 1,200 pixel resolutions. The HDMI standard as used on TVs is also creeping onto some cards, especially those designed for use in home theatre PCs, and includes both digital video and audio signals.
Joining these two well establishing interfaces is DisplayPort. Think of it as a cross between DVI and HDMI and you'll get the idea. It's intended to be more flexible and support higher resolutions than either DVI or HDMI.
Finally, there's the question of support for HDCP encryption (required for Blu-ray playback and other protected content). Most modern cards are HDCP compliant, but it's a feature that's always worth checking.
Form factor and power
Gone are the days of simple, single-slot boards that drop into almost any system. Today's cards vary wildly in size and shape. The biggest boards occupy the space of two PCI Express slots and may be long enough to cause fitting issues in standard ATX chassis.