If the grid fails once or more, then the Power Fail light will be on when grid power returns. While the grid power is off, both lights will be off. The battery used is an ordinary 12 volt car battery. You can buy whatever size battery you feel necessary. You may want to consider using a marine duty battery which will give your more capacity and will hold up better to deep discharging.
Battery capacity is measured in amp hours.
A battery rated at 10 amp hours can supply 1 amp for 10 hours, or 10 amps for 1 hour, or 2 amps for 5 hours, etc. To determine approximately how long you can run your pond equipment on the battery, divide the amp hour rating of the battery by the amps required by the inverter - the result is your run time in hours. If you need more battery capacity, you can put multiple batteries in parallel. If you put batteries in parallel, don't mix and match - only use new and identical batteries. The inverter is used to convert the 12VDC from the battery into 120VAC for use by your pond equipment during power failures.