M68mt-s2 compatible with radeon 7870

Yes... its compatible. Thats a really trashy motherboard though, but thats beside the point. What brand and model of power supply? One of good quality is highly advisable, 500 watt is not sufficient information.

Yes, that mobo is pretty old. I'd upgrade the mobo for better performance and reliability. ANd what PSU do you have?

Well since yesterday, I have been researching and found an Intel® Core™ i7-3770 processor based on a B75MA-P45 (MS-7798 ver 1.0 ) Motherboard. It's in my budget and
Was wondering what type of graphic cards it will accept. There's no details on the PSU and the PC is named 'Advent DT2312' http://www.pcworld.co.uk/gbuk/advent-dt2312-desktop-pc-...

Generally speaking, as long as your motherboard has a PCI-E 16x slot, you will be able to use any type of modern GPU.

On a side note, are you planning to upgrade your CPU and Motherboard? If so, you could invest your money in an i5 3450 and H77M motherboard to give you a bit more flexibility over the B75 option. B75 is really meant for businesses and you may run into some issues with RAM frequency compatibility in the future if you choose to use anything higher than 1333Mhz.