Web hosting is the term applied when someone provides storage space for internet information. This information is usually stored on web pages. A directory (or folder) of web pages is usually called a web site.
If a you wanted to continuously connect your computer to the internet, then you could host your own site. Most people though, prefer to have an internet service provider host their site. Most ISP's have large computer with much, much, much more storage space than the usual home computer. Also most ISP's usually provide a direct internet connection using fiber-optic modems or special hard wire modems that communicate much faster than the common modems in most home computers.
An often overlooked ISP is your internet dial up account. Many dial up account providers such as AOL, Prodigy, Comcast, etc., provide their members with a couple to several megabytes of personal web space. What you can publish in these sites is limited to the agreements that each provides with their services. If you are new to web publishing this is often the best place to publish your first site since these companies usually provide you with a lot of information on how to get that first page published. Check out their members information (services) area for more details.
Things to check for when choosing an ISP:
See my related links page for some of the internet service providers that I've used in the past.