There has been a lot of reference to how one can use a remote Monero node for transactions. So what actually is a remote node?
The simplest way to explain what a remote node is would be to say that it's any node that is not local.
If you are running a node at your home or place or business, you are running a local node. Calling a node a "local" node, means you access it locally. This means you're not going out on the internet to access it. Rather, the node is available on your local network.
Let's say you're a business owner, and you run a node from your office. When you're at work, that node is considered to be a local node, from your perspective. You can hop on your work computer and log into the node on it's local IP address (often 192.168.x.x or 10.10.x.x). But maybe sometimes when you're home at night you'd like to access that node. From that perspective, the node is a remote node. Whenever you're connection is leaving the safety of your local network and going out over the internet, that means you're accessing the node remotely, which means it's a remote node, from that perspective.
Generally, however, people tend to refer to remote nodes as only those remote nodes which they don't own. So when someone says that you're compromising your privacy by accessing a remote node, especially to create a transaction, that's what they mean.
One final point: perhaps you have your own remote node hosted on a VPS. And maybe you set up a VPN between the VPS and your local network. From that perspective, even though it's technically a remote node, the node will appear to be on your local network (assuming the VPN is set up appropriately).
To boil it all down, as already pointed out, (a) a remote node is any node which is not a local node, and (b) calling a node a "remote node" tends to imply that the node is not owned by you.
Since Monero has a separate daemon
monerod, which synchronises with the network, a command line wallet
monero-wallet-cli and a GUI wallet
monero-wallet-gui to generate keys, sign transactions etc.
Both wallet versions need a fully synchronised node to scan for new outputs. For best privacy, it's recommended to run an own node (locally or remotely hosted), since you will not disclose for what outputs your wallet is searching on the blockchain. However, when you have not the resources available due to a slow internet connection or no option to run a node 24/7, you can connect to an open node over the internet, which will save you bandwidth and disc space. For example moneroworld.com offers some open remote nodes.
For broadcasting own transactions you can probably keep even more privacy when using a hidden node over the i2p network, since you will not disclose your IP address along with your transaction to anybody.