A node is any computer running Bitcoin software, such as Bitcoin Core, that connects to the network. Nodes are the network. There is no central server underneath them.
What a node does
A node receives transactions and blocks from its peers, checks them against the rules, and passes the valid ones on. It keeps a mempool of unconfirmed transactions and a copy of the blockchain. If a block breaks a rule, for example by creating too many coins or spending a coin twice, the node rejects it, no matter who mined it.
Full and lightweight
A full node verifies every block from the genesis block onward. Lightweight clients, often called SPV wallets, download only block headers and trust that the chain with the most work is valid. They are convenient, but they verify less.
Nodes vs. miners
People often mix these up. A miner proposes new blocks, and nodes decide whether to accept them. If miners produce blocks that break the rules, nodes simply ignore them, and the miners waste their electricity. This balance is a big part of how Bitcoin reaches consensus without anyone in charge.
Why run one?
Running your own node means you do not have to trust anyone else's view of the ledger. Your wallet can ask your node about your balance instead of a third-party server, which also improves privacy. The software is free and runs on a modest computer or a small single-board device with an external drive. Tens of thousands of people around the world do this, and every one of them makes the peer-to-peer network harder to censor or shut down.
In short: miners write, nodes check. As long as many independent nodes check, the rules stay the rules.