A 'full node' refers to a fully validating node on the bitcoin network, meaning the node validates each block and transaction that happens on the network. An example of this is bitcoin-core.
A 'lightning node' refers to a node on the lightning network, which is a network that exists on top of the bitcoin network. The lightning network has it's own protocol for communicating lightning transactions, and the lightning node will follow that protocol in order to interact with the network. An example of this is lnd.
If a lightning node, that is not a full node, opens a payment channel with another lightning node, how can the first peer check that the other peer does not send a "malicious" closing-channel transaction to the blockhain?
A lightning node will interact with a bitcoin full node in order to watch the chain for such transactions. Since the lightning network operates on top of the bitcoin network, the lightning node will need information from the bitcoin network in order to function properly.