10.6.3 would be your first target, then update (Free) to 10.6.8. But you don't say what model you have.
Check out via Apple menu > About this Mac > More info. Post back with model details but not serial number.
You can install Snow Leopard, Lion or Mountain Lion.
Mountain Lion will be released next week, so upgrade to Snow Leopard and buy it when it is launched. You will keep everything if you upgrade (do a backup before this)