An IP address is a unique identifier for a device on a network. It's used to route traffic to the correct device and can be thought of as the device's address on the network. Just like you need an address to send mail to someone, devices on a network need an address in order to communicate with each other.
IP addresses are made up of four numbers separated by periods (also called octets), with each number ranging from 0-255. For example, an IP address might look like this: 192.168.1.1. You'll sometimes see people refer to this type of IP address as a dotted quad.
Each device on your home network has its own unique IP address that other devices use to identify it on the network. Your router also has its own IP address that's used to identify it on the internet; this is called your public IP address. When you connect to the internet, your ISP assigns your router a public IP address which is then used to route traffic from the internet to your home network.