DA or "Dual Action" is a predictable up-down and back-forth motion. Random action orbital is just as the name implies, a series of random movements in a circular pattern.

Electric is good. Most come with Velcro type pads in grits from 60 - 220. DeWalt, Ryobi, Makida on the less expensive end. Porter Cable and Milwaukee on the more expensive side.

For really super fine finish, Dynabrade or Hutchins (air driven) but they are expensive. Dynabrade 6" is $325+/-. Hutchins 6" will run $225. These are really good units but require a good flow of air (10CFM @ 90psi)