A node is any computer that is connected to a cryptocurrency network, for example, any computer connected to the Bitcoin network can be regarded as a Bitcoin node. Depending on its type, a node can store some or all of the blockchain data.
A full node is any computer connected to the blockchain network 24/7 and fully synchronized with it. Full nodes store all data of the blockchain starting from the Genesis block.
Full nodes serve the network free of charge, they download and validate every block and transaction according to the rules of the consensus algorithm. If any block or transaction disobeys these consensus rules, then they are rejected by the full node, and not included in the blockchain.
A lightweight node is any computer with special software connected to the blockchain network. A lightweight node needs only download the headers of all blocks on the blockchain, which means that download and storage requirements are significantly less intensive than that of a full node.
It is also important to note that lightweight nodes are effectively placing their trust in full nodes in ensuring that blocks and transactions are being correctly validated against consensus rules.