What is the difference between public ip and private ip?

Private IP addresses exist to save on the limited number of available public IP addresses. There are actually 4 ranges of private addresses: 10.0.0.0 to 10.255.255.255 when you need a lot (16,000,000+) of addresses in your internal network; 172.16.0.0 to 172.31.255.255 when you need up to a million or so; and 192.168.0.0 to 192.168.255.255 or 169.254.0.0 to 169.254.255.255 when you only need up to 254 addresses. These addresses must exist behind a router or gateway, not on the live Internet.

The real genius of these private IPs is that you can get one IP from your ISP (let's say 205.20.2.105), and your router / gateway basically splits that into however many internal (private) addresses you need.

I was going to say, but I see the guy below has it very well covered. No use sayuing the same thing he did.

When you're hooked up to a router, the router gets one IP address from your ISP, then it arbitrarily assigns IP addresses (usually 192.168.x.x) to whatever's hooked up to it. The router's address is your public IP, and the IP assigned to your computer is your private IP.

When you, say, request a page from your computer, the request carries the public IP. When the router receives a response, it forwards it to your private IP. You can also use private IPs to communicate with other computers on the same router.

A public IP address makes your equipment accessible to everyone on the internet and is needed for VoIP or if you want to give others to access to specific equipment on your network. You must therefore be careful to protect your equipment from hackers and viruses and not to allow your equipment to be hi-jacked and used as an open email-relay, for example. A private IP address is for private use within the network and allows many more PCs to be connected. If a customer is using a private IP and later wants VOIP they would need to change to a public IP address.
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At a more fundamental level, public ips are 'internet accessible' while private ips are not (at least not directly).