An Internet service provider, or ISP, maintains, installs and provides internet connection to residential or commercial areas. Some examples of ISP clients include - homes, coffee shops, hotels, libraries, offices. They usually charge a fee for installing the connection and a monthly fee for maintaining it.
The internet is basically a web of interconnected networks - which are maintained by a huge number of different ISPs. They keep connected around the world by giving their clients access to other ISP networks. This is called peering.