A "node" in the Bitcoin network (or any other P2P cryptocurrency, like Litecoin, which is based on Bitcoin), is a software application running on a public facing computer or server. This software connects and interacts with other Bitcoin nodes, to form the Bitcoin P2P network. In simple words, the Bitcoin network consists of computers running Bitcoin node software. Without nodes, there would be no Bitcoin.
Nodes relay to each other transactions and newly-mined blocks, as well as the IP addresses of other nodes.
Some nodes maintain a full copy of the huge Bitcoin transaction database (called "blockchain") in order to validate every new transaction and block before relaying them to other nodes. These nodes are called "full nodes".
Running a full node on your computer or connecting directly to a full node means you don't have to rely on any third-party to "process" and validate your bitcoin transactions. Therefore, full nodes are essential for the integrity of the blockchain.
Full nodes usually accept incoming connections from other nodes, using port 8333. It is necessary for Bitcoin network to have adequate number of full nodes that accept incoming connections.