When you send any kind of message (e.g. a letter or text message), you have to define the recipient or destination address. Of course, you are the sender and your address is the sender’s address. In case of computer networks the data are transmitted as packets and every IP packet has to have the source and destination IP addresses. Therefore, each computer and server connected to the Internet network has its unique IP address.
The IP network is a network which uses the IP (Internet Protocol) for communication between all elements. The data between hosts (e.g. computers) are transmitted via one of the transport protocols: TCP (Transmission Control Protocol) or UDP (User Datagram Protocol). The TCP is used by most of Internet applications like web browsers, FTP, p2p and e-mail clients. The UDP is faster that the TCP, but it does not guarantee that the data will not be lost during transmission. Therefore, the UDP is used for real-time services like VoIP (Voice over IP) and on-line gaming.
We have a lot of devices connected to the Internet and limited amount of IP addresses. Therefore, it is not possible to assign the unique address to each device. This is the reason why private and public IP address pools have been defined. For an example, your computer has IP 192.168.0.102 and the same IP can be used in other local area networks. However, all computers from your home network are seen on the Internet at a different IP address which is unique and it can not be reused by other network. This is the public IP address.
In huge networks (e.g. like Internet), computers are not connected directly with each other and we have nodes between them. These interconnection nodes are called routers and they decide which path will be used for transmission from point A (source IP address) to point B (destination IP address). Of course, the Internet is protected against a failure of a singe path, so there is several transmission paths. The router thanks to dynamic routing protocols is able to choice the shortest and most reliable of them.