The Internet is the set of computers that that talk to each other using Internet Protocol.
Internet Protocol specifies a data packet transport system. It was designed for use on battlefields by researchers working for the U.S. Department of Defense. Because of the design, there is no one organization that controls the Internet.
Many protocols run on top of Internet Protocol. Among them are File Transfer Protocol (FTP) and Hyper-Text Transfer Protocol (HTTP). Most of the World Wide Web is distributed using HTTP.
There are several levels of connecting to the Internet: your computer can be on the Internet; your computer can access a computer that is on the Internet; or you can use someone else's computer that is on the Internet.
When you use Point to Point Protocol (PPP) your computer is on the Internet. Any program that is designed to access the Internet can run on your computer. Your computer is part of the Internet and other computers can access services that are hosted on your computer.
When dial in and use a shell of some sort, then you are using a computer that is on the Internet. Since that computer is likely to be connected to the Internet 24-hours per day, you can offer services by using such a system.