Batteries are commonly used to power small devices, such as cellphones, flashlights, watches, etc. These batteries generate electricity by chemical reactions. As the reaction progresses, electricity is produced and this, in turn, powers the devices. Some of these reactions are reversible, while others are not.
The batteries that run on non-reversible chemical reactions are known as non-rechargeable batteries or primary batteries. The batteries that operate on reversible reactions are known as rechargeable batteries or secondary batteries.
The concept of rechargeable batteries is that the chemical reaction, that supplies electricity, can be reversed by supplying electricity from an external source, thus recharging the battery. Some examples of rechargeable batteries are Ni-Cd and Lead-acid batteries. Rechargeable batteries have made the large scale adoption of cellphones possible. Rechargeable batteries are more expensive to buy than non-rechargeable batteries.
Note that rechargeable batteries have a fixed number of recyclings possible and hence can not be recharged again and again indefinitely. Non-rechargeable batteries have to be disposed off after a single use (once they are drained of all the charge) and hence require frequent replacement.
Hope this helps.