Battery: Battery warning light is on

What you are seeing is actually the warning indicator for the battery charging system, rather than the battery itself. When you first switch the ignition on, you should see that sign come up, then when the engine starts, its should extinguish within a few moments.

The repair you will most likely require is an alternator replacement or alternator drive belt replacement. In most cases you can easily see if the belt has come off as there will be pulleys on various parts of the engine (obviously the alternator being one of them!) that are no longer being turned by anything.

Now the technical waffle!
There is a lot of misunderstanding regarding how a vehicles electrics are powered, to help clear this up, basically when the engine is running, the majority of the electrics are run from the alternator (which is the part that also charges the battery)

If the light stays on, or comes on when you are driving, it means the vehicle is no longer working from the alternator but is using the charge that is left in the battery. This will quickly dissipate and certain electrical functions will shut down, usually ABS (you will see a red ABS warning on the dash), PAS (if you have electric steering) and eventually your engine will cut out. You are now stranded!

Therefore, if you battery light DOESN'T go out, don't risk the journey no matter how short.