Replacing a battery is very easy. Just remove any clips that hold the old one in position and whip it out (to your local recycling centre!). Install the new one, reattach the cables to the correct sides and go.
Have a look at the old battery to find out its capacity. It's almost definately 12 volt, but you need to know its amperage. Somewhere on a label, it'll say something like 60Ah (Ampere-hours). You'll need the new one to be at least the same as your current one. Bigger ampere-hours is ok, but make sure the battery will physically fit in the space available.
Alternatively, go to Halfords or the like, where you just look up a list and it'll tell you which battery to buy.
The engine "turning over" just means it is rotating powered by the starter motor.