VA = the power rating of the UPS as a product, i.e. its total peak (reactive) power capability.
VA = Volts (potential) * Amps (current) = roughly equivalent to Watts in a resistive load, though when you're powering items uncorrected for power factor like some motors or less quality PC power supplies the VA used is higher than the Watts used by several percent or sometimes quite a bit more.
WH = Watt Hours = how long in hours a UPS can run a device that takes one watt. Since W is approximately equal to VA, WH = a bit less than VAH.
AH = a battery rating of AMP HOURS which indicates how many AMPS current the battery can put out for one hour before the battery is effectively discharged. Typically you don't want to discharge a battery much if you can help it, since the product lifetime and future capacity often dramatically decreases dramatically with how many times you significantly discharge it. Ideally you stop a discharge when it is still around 70% capacity... Draining it all the way down is not suggested.
So a 12V 20AH battery can possibly hypothetically supply somewhere under 12V at 10A for somewhere around 2 hours, 10A * 2H = 20AH, and 12V * 10A = 120VA or around 120W.
A typical UPS is not designed for sustained long term backup operation. Typically at 80% of their rated VA output level many consumer units only have battery power capacity (energy) to last approximately 3-5 minutes before the battery is totally dead (not a good thing).
So a 1000VA (1000W) unit capable of running 6 minutes (1/10 hour) would have a battery capacity of around 100VAH or 100WH. Since the batteries are usually 12V units, 100VAH translates into a 12V 8.3AH battery being discharged at 83A out of the battery to generate 120V UPS output power at 8.3A with a run-time until discharge of 1/10th hour.
If your 1000VA UPS had a bigger (or infinite) battery it'd be happy to keep supplying 1000VA forever or for longer than 3-6 minutes anyway. As it is, it stops when the battery dies. Put in a smaller battery, you get out 1000VA for less time until the battery dies. Put in a bigger battery and you get 1000VA for more time until the battery dies.
As to whether you can successfully install and use a larger capacity battery than the UPS' original one, that depends on whether the UPS has internal charging and run-time cut-off logic that just "assumes" the battery has a certain capacity and only charges / discharges it to that level, or whether it detects the actual charge state and charges / discharges accordingly.
Also physically larger batteries with more AH capacity at the same voltage may not physically fit into your UPS. And some batteries may not be rated for the same maximum discharge or charge current or number of charge/discharge (lifetime) cycles as others, so that also complicates the decision.
In general, though, when you can get away with it, the bigger the battery capacity used in the UPS the longer the run-time "on battery" may be or the longer lifetime (until it needs replacing) the battery will give since you're using it less severely in terms of discharge depth for a given WH discharge level.