In more detail
IPv4 addresses are 32 bits, written as four numbers from 0 to 255 such as 192.168.1.10. IPv6 addresses are 128 bits, written in hexadecimal. Addresses in the private ranges (10.0.0.0/8, 172.16.0.0/12, 192.168.0.0/16) work only inside a local network, while public addresses are reachable on the internet. A subnet mask splits an address into network and host parts. Devices get their address from DHCP or manual configuration; public ranges are allocated by regional registries to providers.
Why it matters
Firewall rules, server logs, abuse reports and geolocation all work with IP addresses. Because of NAT, websites see your router's public address, not your device's private one, and geolocation from an address is only approximate, often just the provider's region.
Example
Your home devices may be 192.168.1.x, while the whole house appears to websites as a single public address. Look up yours with the What's My IP?. Underneath, an IPv4 address is just a 32-bit number: 192.0.2.10 is 192 × 2^24 + 0 × 2^16 + 2 × 2^8 + 10 = 3,221,225,994. The blocks 192.0.2.0/24, 198.51.100.0/24 and 203.0.113.0/24 are reserved for documentation and examples.
Common confusion
A MAC address identifies a network interface on the local link; the IP address is what routes traffic across networks.