Your question is unusual because most sinks before the introduction and wide use of single-handle faucets had a separate valve for the hot and and another for the cold side. (Before that, there was only one handle for cold water; the hot water came from the stove.
Whether a single handle or two separate ones, the cold is connected to a cold line and the hot is connected to a hot water line. They can be connected by metal or plastic tubing. There are flexible hoses covered by metal braid available also; some single-handle faucets come with the flexible connectors in the box.