The amount of electrical power a battery can deliver is the maximum rate at which energy from the battery can be safely discharged, known as the discharge power capability, it is given by the 'E-rate' of the battery. For example, the E/10 rate for a cell or battery rated at 173 watt-hours is 1.
What is the relationship between power and battery capacity?
The higher the power, the quicker the rate at which a battery can do work—this relationship shows how voltage and current are both important for working out what a battery is suitable for. Capacity = the power of the battery as a function of time, which is used to describe the length of time a battery will be able to power a device.
The higher the current, the more work it can do at the same voltage. Power = voltage x current. The higher the power, the quicker the rate at which a battery can do work—this relationship shows how voltage and current are both important for working out what a battery is suitable for.
This is not possible. Aside from the fact that batteries can not provide infinite current (they have internal impedance), to supply infinite current requires infinite power, to supply infinite power for any duration at all requires infinite energy. Incidentally, batteries do not contain infinite energy.
How is a battery rated?
Typically a battery is rated for power with something called a "C" rating, or how much power it would take to drain the battery in one hour. Since output power of a battery is voltage times current, the C rating can be calculated as nominal voltage times the amp-hour rating, divided by the nominal voltage times an hour.
With a battery, generally the higher the energy density the better, as it means the battery can be smaller and more compact, which is always a plus when you need it to power something you want to keep in your pocket. It's also a plus for electric cars—the batteries have to fit in the car somehow!
Neither, it means more energy and it implies more power. Think of energy as the thing you "spend" to do work, and power is how much work you get done in a particular period of time. Typically a battery is rated for power with something called a "C" rating, or how much power it would take to drain the battery in one hour.