There are two fundamental influences on battery capacity: rate of discharge and temperature. Discharging a battery quickly will reduce its capacity, as will using it in cold temperatures. Discharging a battery slowly, or in warm temperatures will increase its capacity.
Batteries are rated in their Amp-Hour (AH) capacity. An ampere is a measure of electrical charge flowing across a point in a circuit for a unit of time. Higher electrical charge = higher current.
Amp-hours are Amps x hours (time).
10AH = 10 amps for 1 hour / 1 amp for 10 hours
35AH = 35 amps for 1 hour / 7 amps for 5 hours
200AH = 200 amps for 1 hour / 10 amps for 20 hours
However, it isn't as simple as this. If the battery is rated at 200Ah for 20 hours yet you discharge it over 100 hours, you will get more than 200Ah of power.
High use, low capacity
Due to internal resistance within a battery, the capacity of a battery changes depending on the way it is being used. Higher discharge increases resistance and reduces capacity. Lower discharge reduces resistance and increases capacity. Treat your battery good, and it'll treat you better.
Often, batteries are listed with their 20 hour rate (C/20). A more aggressive discharge may be a 10 hour rate (C/10) and a much more gentle discharge may be a 100 hour rate (C/100). You will often find these three figures on a battery listing.
The graph shows three relatively similar deep-cycle lead-acid 2V batteries. On all three, the capacity increases with the slower discharge rate of C/100. Internal resistance is low and less capacity is lost as heat. The rapid 10 hour discharge rate sees capacity drop by 1000 Ah on all three batteries.
The amp-hour capacity of a battery is heavily affected by temperature. Standard capacity ratings are at 25C - room temperature. Capacity increases at higher temperatures, up to 120% at 50C. It decreases by about 50% when at -20C. At freezing, capacity is around 70%.
High temperatures lower internal resistance and increase the speed of the chemical reaction. Colder temperatures do the opposite and slow the chemical reaction. When considering the storage and operating temperature of a battery, the trade-off between short-term capacity and long-term lifespan must be a factor. Operating at high temperatures may provide immediate power boosts, but the as the old adage goes, the tortoise wins the race. The drop-off in battery lifespan at high temperatures must be treated with care.
Operating at extremely low (freezing) temperatures, will limit the capacity as the electrolyte freezes after a certain DoD. This will severely damage the battery.