The main difference between public and private networks, apart from the fact that access to a private network is tightly controlled and access to a public network is not, is that the addressing of devices on a public network must be considered carefully, whereas addressing on a private network has a little more latitude.
As already discussed, in order for hosts on a network to communicate by using TCP/IP, they must have unique addresses. This number defines the logical network each host belongs to and the host's address on that network. On a private network with, say, three logical networks and 100 nodes on each network, addressing is not a particularly complex task. On a network on the scale of the Internet, however, addressing is very complex.
If you are connecting a system to the Internet, you need to get a valid registered IP address. Most commonly, you would obtain this address from your ISP. Alternatively, for example, if you wanted a large number of addresses, you could contact the organization responsible for address assignment in your geographical area. You can determine who the regional numbers authority for your area is by visiting the IANA website.
Because of the nature of their business, ISPs have large blocks of IP addresses that they can assign to their clients. If you need a registered IP address, getting one from an ISP will almost certainly be a simpler process than going through a regional numbers authority. Some ISPs' plans actually include blocks of registered IP addresses, working on the principle that businesses are going to want some kind of permanent presence on the Internet. Of course, if you discontinue your service with the ISP, you will no longer be able to use the IP address they provided.