like the other's have said, more info is needed. If you don't know the specs off the top of your head, windows has a utility called DxDiag.exe. You can find it by searching for it from the start menu.
Open it, and it'll show you the System tab. If you could screenshot the System and the Display tabs and post them here, it will give us the information we need.
example of display tab on my computer:
(note I'm running windows in a VM so the data you see will be a bit different on your machine)
That said, I speculate that a better graphics card would be the first upgrade to make. Im assuming this machine was bought prebuilt (think Dell/HP/etc).OEMs typically cheap out on the GPU, so even though it has 1GB of memory, it likely doesn't have the processing power to run games well.
Upgrading the GPU gets a bit tricky on prebuilt computers for a number of reasons. The most obvious is space. High performance graphics cards (GPUs) are generally rather large, and most OEMs use small cases so making the card fit can sometimes be a problem. If you have a normal sized tower then you're probably ok here. If its a slim style case then you're kinda out of luck short of buying a larger case and moving everything over.
Second power can be an issue. Dell/HP/etc like to skimp here too. Most of the time they use a power supply rated for <300 watts. Top of the line GPUs can easily draw more than that. Luckily recent mid range cards have become very power efficient so you could potentially be ok there too. Additionally higher power cards will require supplemental power connections from the PSU, which yours likely doesn't have. There are decent cards that don't need supplemental power though, so this may or may not be a problem.
Third is available PCI-E slots.
PCI-E is the standard that almost all desktop GPUs connect to the computer with. Most of the time OEMs will include at least 1 full length slot (referred to as 16x since there are 16 PCIE lanes available). If you have such a slot then you're probably good. If not, you'd have to buy a different motherboard and at that point you're probably better off just saving up for an entirely new system.
Hopefully that makes sense.