Quick graphics card question?

yes, any would do, just be sure your power supply can handle it.

which it can't.

upgrade to a 500-600W power supply.

if you use it on that build right now, card might not even function properly to the lack of power.

So just get a new power supply., nothing else :)

also, motherboards always have a pci express X.0 x16 slot which fits all video cards, since manufacturers made them backwards compatible to easily upgrade to a new one.

500W (or greater) power supply with one 75W 6-pin PCI Express power connector recommended

Yes, and your power supply should be more than adequate.

Also, make sure you have enough space in your case. Its lenght is 210 mm.

computer + AMD Radeon HD 7770 at 55 watts = 259 watts if your PC is from HP, i was told not all powersupplies will fit in the HP case, but i don't know if that is true

Just make sure you have the right slot on your motherboard, the card will fit, and I recommend upgrading your PSU to 500-600 Watts. I have never heard of a modern computer with a decent GPU and CPU running on a PSU with less than 400 Watts. The 500-600 Watt PSU will also last for awhile and give you plenty of room to XFire or upgrade your CPU.
A new PSU not pushed to the limits will also allow you to install more fans and other things that will use some of those Watts, which on your current CPU, even if everything ran, wouldn't have room for anything else.

Your PC description tells us nothing about motherboard slots or case size which is important thing here. Also, is that card slot powered, or does it require extra supply and if so does your PSU have the right cable?

Yes, it can.
Afaik Radeon 4000 series were built for pci-e 2.0 and that would mean that a 7770 would fit too.