A server is a powerful computer that stays switched on all the time and whose job is to store websites and hand their pages to visitors the moment they are asked for. When you open a website, a server somewhere is quietly sending it to you.
Picture the kitchen of a busy restaurant: guests (visitors) place orders, and the kitchen (the server) prepares and sends out each dish (web page) as requested, all day and all night. A good kitchen serves many guests quickly without mistakes — and a good server does the same for a website.
Renting space and power on such a computer is exactly what Hosting provides. In other words, hosting is the service, and the server is the actual machine doing the work behind it.