Class A - 255.0.0.0 - 11111111.00000000.00000000.00000000
Class B - 255.255.0.0 - 11111111.11111111.00000000.00000000
Class C - 255.255.255.0 - 11111111.11111111.11111111.00000000
CIDR -- Classless InterDomain Routing.
CIDR was invented several years ago to keep the internet from running out of IP addresses. The "classful" system of allocating IP addresses can be very wasteful; anyone who could reasonably show a need for more that 254 host addresses was given a Class B address block of 65533 host addresses. Even more wasteful were companies and organizations that were allocated Class A address blocks, which contain over 16 Million host addresses! Only a tiny percentage of the allocated Class A and Class B address space has ever been actually assigned to a host computer on the Internet.
People realized that addresses could be conserved if the class system was eliminated. By accurately allocating only the amount of address space that was actually needed, the address space crisis could be avoided for many years. This was first proposed in 1992 as a scheme called Supernetting.
The use of a CIDR notated address is the same as for a Classful address. Classful addresses can easily be written in CIDR notation (Class A = /8, Class B = /16, and Class C = /24)
It is currently almost impossible for an individual or company to be allocated their own IP address blocks. You will simply be told to get them from your ISP. The reason for this is the ever-growing size of the internet routing table. Just 5 years ago, there were less than 5000 network routes in the entire Internet. Today, there are over 90,000. Using CIDR, the biggest ISPs are allocated large chunks of address space (usually with a subnet mask of /19 or even smaller); the ISP's customers (often other, smaller ISPs) are then allocated networks from the big ISP's pool. That way, all the big ISP's customers (and their customers, and so on) are accessible via 1 network route on the Internet.
It is expected that CIDR will keep the Internet happily in IP addresses for the next few years at least. After that, IPv6, with 128 bit addresses, will be needed. Under IPv6, even sloppy address allocation would comfortably allow a billion unique IP addresses for every person on earth