Well you'll need to look inside your computer first. Here is a picture for reference:
Just unscrew the side panel from your case, take it off, and check for those slots.
Your power supply is also important. It will determine what sort of card you can use. It's a box with wires coming out of it. The most important things there are
12V Wattage and Amperage which will be listed on a sticker on the side of the PSU (power supply) - Note what it says and post it here. Beefier graphics cards require more power.
Furthermore, the amount of memory on a graphics card will tell you nothing of its performance! Things to look for include:
Stream Processors - Higher is better. Nvidia and AMD use different types. So an Nvidia card with 384 stream processors might be better than an AMD card with 800 stream processors. But it's still useful for comparing the same brand of cards
Memory bandwidth - Again, higher is better. Most cards come with 128-bit, 256-bit, 320-bit or such. A higher bit memory means more bandwidth.
Card series - For example, GTX 400 series, GTX 500 series, Radeon 5000 series, Radeon 6000 series are the latest series of cards from AMD & Nvidia.
Clocks - You'll see these listed on the box. For example:
Core clock: 675MHz
Memory clock: 4000mhz effective
The "effective" part means that the memory is 1000Mhz or 2000MHz. GDDR5 memory is generally "QDR" or quadruple data rate. GDDR3 is "DDR" or double data rate. What does this mean? It means that the memory can perform more than one thing per clock.
Going back to power supplies. You will likely need a new one for a new graphics card. Just get back to us with the specs.