Whether you want a super high powered Rocketship of an e-bike, or a super casual Sunday cruiser for cruising the Boardwalk, making sure you select the correct battery is important for many reasons. Financially, performance and safety all come into play.
The main numbers you want to pay attention to are Voltage ratings, Amp ratings and Amp Hour ratings.
Matching your motor voltage and your battery voltage cannot be understated if you want your setup to even work, let alone cause serious damage. If your motor is rated at 36v, get a 36v battery and so on. Getting a 72v battery and a 48v motor will likely fry your electronics located in the motors controller. Using too low of a voltage will not give enough voltage to even register in the controller and you will not be able to power it up.
Some motors have a variable voltage they can run off and are usually clearly marked. Such motors, like the CYC X1 Pro can use any battery from 36-72v.
If your motor is rated as a 48v, it can almost be assumed that a 52v battery will also work as the controller reads them as the same for the most part. In an effort to keep this article short, we will leave it at that :)
Amps, or, current is the amount of energy, or watts flowing through the system. A good analogy is thinking about water pressure coming out of a garden hose, the more pressure you allow into the hose by opening the valve creates more water at the end of the hose. small pressure creates a small dribble of water and fully opened valve creates water shooting out. More power.
Every motor has a different maximum amount of amps it is capable of drawing from a battery. You don't need to worry about getting a battery with an "AMP" rating higher than your motors "AMP" rating, you actually want that. Getting a battery that has a lower "AMP" rating than what the motor needs will create a lack of performance at best and a safety issue at worse. I cannot overstate this enough, if you motor draws 25a max, get a battery that can output at least 25a continuously, a higher rating is ideal as you wont be pushing the batteries BMS (Battery Management System aka: brains) to its limit. If you are unsure on your motors amp draw, ask!
This refers to how many specific amps the battery can output for an hour. Say you have a motor drawing 25 amps from the motor continuously without stopping for an hour exactly before the battery dies would mean it can output 25 amps for an hour, or 25 amp hours (25ah for short). This rating is related to range. If you have a 12.5ah battery, using the above example, would net you half the range as it would only be able to output 25 amps for half an hour.
I cannot mention this enough. Combining batteries can be very dangerous if you do it incorrectly! Never simply combine wires, there are numerous examples of fires starting that way. Always use a battery balancer/ combiner that was designed to combine batteries safely and always use the same voltage batteries.
Combining batteries doesnt add voltage or power to the system, it adds the Amp Hour (ah) rating of each battery to give you more range only.
In Summary, it almost never hurts to go with an "Overkill" battery, meaning it is of much higher performance ratings than the motor requires. In fact I always recommend making sure the battery is at least a little bit "Overkill"
However, if your setup doesn't require high Amp or ah ratings, you can save a lot of money by getting a high quality battery built with lower grade components inside. Its like buying a Ferrari to go get groceries. With that being said, be sure to do your Due Diligence and making sure your battery was assembled using high quality BMS boards and other materials as cutting corners on materials to keep battery prices low can cost you dearly.