IPv6
Because there are no more unassigned IPv4 addresses, the world is in the process of switching over to a new 128-bit IP system for the Internet, called IPv6. This system includes six separate numbers (rather than four), and each of those numbers can be as high as 65,535 (216). This system provides 2128 (more than 340 trillion, trillion, trillion) possible addresses, making it likely that there will be enough addresses available for the foreseeable future.
The transition to IPv6 has been much slower than it was expected to be, due in part to existing systems that revolve around IPv4 and that cannot easily adapt to the new system. It is also quite likely that people will use IPv4 for internal network addresses for many more years, even after the Internet has officially transitioned to IPv6.
For more information about IPv6, see the Internet Society's explanation at IPv6.