This is Remix’s own internal sandbox. It does not connect to MainNet, TestNet, or any private network. It’s an in-memory blockchain that can be used for simple testing and quick mining.
This is the option to use for the browser plugin (MetaMask). Here, you are telling Remix to defer all control of the blockchain integration to the MetaMask plugin. At this point, MetaMask controls which network you are connecting to.
In the plugin, you can connect to MainNet, Ropsten, Rinkeby, etc through Infura’s node network. In this case, you’re not running a node locally. MetaMask also has a localhost option where you can run your own node locally and MetaMask will send all transactions to it (this local network can be a private network using any node client or you can use a test blockchain like TestRPC).
This allows you to enter a URL in Remix to connect to the blockchain. The most common setup here is to be running a local node and connecting to it through it’s IP/port.
This is pretty much the same as using MetaMask’s localhost option, but you’re just cutting the plugin out of being the middleman. Just like option #2, the network your connected to depends on how you’ve configured your local node (it can be main, test, private, etc.).