The ECS 945GCT-M2 has one PCIeX16 slot so the card will work. You will need also 450W PSU with 26A on the 12V rails. As for games, the 3870 can play most games from the time that Bioshock came out till Resident Evil 5 at high settings (with core 2 duo CPU and 1400x900 resolution). Today many newer games require at least 4870 1GB to run at high settings and in a few months many games will also have DX11. Do you have another MoBo option? the ECS 945GCT-M2 has only 2GB RAM support.
Yes it will run as for performance they are a decent card for now however for high ress which is standard these days they are anemic. I own and use a Diamond 3870 and it pulls its weight till I begin to multy box wow only to get hammered.