A website hosting company is one that holds your site in cyberspace. Your website gets loaded onto the hosts computer which is open 24 hours per day 7 days a week. That computer serves the information (i.e., your webpages and other relevant documents) to who ever visits your website (hence the common name "server").
If you want to have a website, you need to get it hosted somewhere. A hosting company (or server) is a company that holds your website in cyberspace, leaving it open for the world to access 24 hours/day.