Batteries are made up of a number of cells, typically around 50.

Cells are normally Lithium Ion '18650s' which refers to their physical dimensions and shape. (See picture)

The orientation and connection of the cells determines the overall voltage of the battery. 

The amount of power (Watt Hours) is denoted by the type and number of cells. Volts x ampHours = Watts Hours. A 100 Watt Hour battery can produce 100 Watts for 1 hour (at a given current output).

Watt hours is the only true measure of energy stored. Sometimes batteries are classed in amps, or amp hours. This is not the whole picture, a high amp battery with low voltage will hold less than a medium amp battery with high voltage.

All decent batteries have a BMS or Battery Management System. This includes temperature sensors to ensure the battery never overcharges, overheats, or is charged in too cold a temperature leading to damage.

One more thing - we still see some eBikes using Lead Acid batteries (like your car). They're not adequate for the task - avoid.

Additional learning sources:

Batteries are the single most important component of any eBike. They are also the most expensive. And the most dangerous.

A top of the range battery from the likes of Bosch or Brose costs upwards of $950.

The only part of an eBike that could cause a fire is the battery. Quality batteries are extremely safe! 
Cheap knock offs are not....

On the left you can see batteries from Bosch (500 Watt Hour) and Brose (650 Watt Hour).

The Brose weighs 7 lbs the Bosch 5 3/4.
18650 eBike cells
But potential energy stored is not the whole picture. A battery can be asked to fill up to 100% and empty to 0%. This obviously maximizes the usable energy but it reduces the number of cycles the battery can perform. Or a battery can be set to charge up to 95% capacity and empty to 5%; this allows the battery to perform many more cycles.
Bosch quotes a 2 year warranty on its 500WH battery and 500+ cycles. 
Brose quotes a 4 year warranty on its 650 WH battery and 900+ cycles.
Of course all batteries die after 5 or 6 that lifetime is only relevant if you are using it a lot! 500 cycles is 2 x 40 mile rides per week over 5 years!
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