The short answer, according to the site How Old is the Internet, is that the Internet is 25 years and 11 months old.

The longer answer is not as simple. The ARPAnet, the predecessor of the Internet, was born in November 1969, making the Internet 39 years old.

In January 1983, ARPAnet shifted to the TCP/IP protocol, which to this date powers the modern Internet. If that is taken as the birth date, the Internet becomes around 25 years old. [via Reddit]