Well, you will need a battery the correct size to fit into your battery hole. Measure very, very carefully.
The battery will have some sort of red and black terminals or wires, these will connect "directly" to the controller. There may, or may not, be some additional wires and/or connectors in-between, which are not necessarily needed.
There are various standard types of connectors, Andersons, Deans, several others. There are also lots and lots of individual and unique ones for various types of integrated batteries.
To find an exact replacement will limit your options, most likely.
If you can fit the size and shape onto your bike, somewhere, it is only a matter of connecting two wires, or splicing on four connectors.
I think you have a vertical seatpost-mount unit meant to remove easily, just do a search on amazon for yukon battery.