In this context, a node refers to a computer that runs a Monero daemon. A node keeps a copy of the Monero blockchain. Your wallet needs to connect to a node to obtain information about your transactions. It does this by scanning the blockchain for transactions that belong to you. Unlike in Bitcoin, the Monero blockchain is encrypted. As a result a remote node can't simply provide you with your balance and your past transactions. Instead, a node provides your wallet with enough information about all transactions on the blockchain for your wallet to determine which transactions belong to you.
A daemon is a piece of software that runs in the background of your computer. The Monero daemon keeps the node in sync with the network and talks to your wallet.
A local node is a Monero daemon that is run on your computer. The blockchain is stored on your own harddrive and your node talks directly to other nodes to keep itself synchronized with the network. A wallet connected to a local node will talk directly to the daemon on your computer to obtain information about the blockchain.
A remote node is node run on a computer controlled by someone else.
Besides local and remote nodes, there exist a non-mutually exclusive subclass of nodes called 'full' and 'pruned' nodes. A full node is a node that keeps the entire Monero blockchain. A pruned node is a new type of node introduced in v0.14.1 that only stores 1/8th of the blockchain, saving space.
There are many reasons why you wouldn't want to run a local node on Tails:
The possible privacy implications of using a remote node over Tor, especially when you connect to a node run by trusted members of the community do not outweigh the effort it takes to set up a local node (in my opinion).
The concern about linking your IP to your transactions when connected to a remote node does not hold if you connect to it over Tor. Exit nodes cannot spy on your traffic if you connect to a .onion remote node. Furthermore, a remote node does not know which transactions belong to you. It does not know your XMR address. It does not know your private view key (unlike web wallets).
There exists a novel attack that a remote node could attempt to execute, which has a 10% chance of succeeding in revealing the true input of a single transaction. The receiver and the amount of the transaction will still be completely hidden. In failure your wallet will warn you to disconnect from the remote node.
Some metadata is leaked to the remote node, such as which blocks are downloaded and when. This may reveal some information, such as what your restore height could be. However, since you never revealed your IP do you really care?
Watch this Breaking Monero episode on remote nodes for further information.