The things you need to look for are whether the RAM is DDR, DDR2 or DDR3 - given the age of the computer it's probably DDR2, possibly DDR.
The second consideration is the speed of the RAM, measured in MHz - this will be along the lines of 667MHz, or 800MHz/1066MHz and so on.
The RAM you buy needs to be the same type and (preferably) the same speed, to avoid compatibility issues.
You'll need to find out what type of RAM you have (it's DDR, according to your signature).
You'll also need to find out what speed the RAM is - this isn't in your signature. To find out the speed, you can either look in the BIOS, or download CPU-Z (CPUID - System & hardware benchmark, monitoring, reporting
) and look in the Memory tab.
One final consideration is:
a) What the maximum capacity for RAM your motherboard has, in GB. - this will be in your motherboard manual. It's probably more than the 1GB you have now.
b) The number of physical slots your motherboard has. Since you're taking up two slots with the two 512MB sticks, if you only have two slots then you'll need to replace the RAM you have now or you can't upgrade. However if you have more than two slots you will be fine.
Once you've found these out, look on Newegg for RAM with these specs.