No one person invented the Internet as we know it today. However, certain major figures contributed major breakthroughs:
Leonard Kleinrock was the first to publish a paper about the idea of packet switching, which is essential to the Internet. He did so in 1961. Packet switching is the idea that packets of data can be "routed" from one place to another based on address information carried in the data, much like the address on a letter. Packet switching replaces the older concept of "circuit switching," in which an actual electrical circuit is established all the way from the source to the destination. Circuit switching is the idea behind traditional telephone exchanges.
J.C.R. Licklider was the first to describe an Internet-like worldwide network of computers, in 1962. He called it the "Galactic Network."
Larry G. Roberts created the first functioning long-distance computer networks in 1965 and designed the Advanced Research Projects Agency Network (ARPANET), the seed from which the modern Internet grew, in 1966.
Bob Kahn and Vint Cerf invented the Transmission Control Protocol (TCP) which moves data on the modern Internet, in 1972 and 1973.
By 1983, TCP was the standard and ARPANET began to resemble the modern Internet in many respects. The ARPANET itself was taken out of commission in 1990. Most restrictions on commercial Internet traffic ended in 1991, with the last limitations removed in 1995.
For a much more complete history, see the web site of the Internet Society.
Note that the Internet and the World Wide Web are not the same thing. See also: who invented the World Wide Web?, What is the difference between the World Wide Web and the Internet? and See also Hobbes' Internet Timeline for another excellent history of the Internet which includes later important events.
NOTE TO STUDENTS: THE INTERNET WAS NOT INVENTED BY ONE PERSON OR GROUP OF PEOPLE. THERE IS NO ONE INDIVIDUAL RESPONSIBLE. I get a lot of email about this. If I gave you just one name, my answer would be completely wrong. Writing down one name on a test will probably get you a bad grade. If your teacher says that one person invented the Internet, show them this page. And don't forget, the Internet and the World Wide Web are NOT the same thing. See What is the difference between the World Wide Web and the Internet?