A network adapter is also called a network card. This card allows the computer to connect to the network. Without this network card there would be no way to connect the computer to a network.
This network card allows the computer to connect to the network through a connection with the router. The router is responsible for directing traffic on the local network and also to the internet as well as subsequent responses.
The network adapter can also be a substitute for a network card if the device does not have a network card. Network adapters are made to be compliant with both types of networks, the wired and the wireless. There is more variety with the wireless version though, because there are more varieties of wireless networks and in order for it to work with all it must be more robust.
You will mainly only find a wireless network adapter in mobile devices such as laptops. If you own a laptop which you take with you regularly for work purposes your card will either need to be updated on an annual basis, or you should purchase a new computer every couple of years.
The reason for this is that technology and especially wireless technology is changing on a constant basis and your card will eventually become outdated and obsolete. It eventually will not work where it once did.