Access Point Name
The hostname (Internet address) of a gateway that connects phones to the rest of the Internet. Most phones require such an access point (gateway) in order to access the Internet. The name of that access point must be correctly entered in the phone's settings in order for the phone to find it on the network and thus connect to the Internet.
The APN should be the gateway that belongs to the carrier (AKA service provider or mobile network operator) matching the SIM card in the phone. For example, if the phone is using a T-Mobile account, then the APN may be something like "epc.tmobile.com" (example only).
Phones sold by carriers should come with APN and similar settings already configured. The APN should only need to be changed when setting up an unlocked phone with a new carrier. Users should consult their carrier's support web site to find the APN settings for their phone.