Definition
An Internet Protocol address is a numerical label assigned to each device that participates in a network. IPv4 addresses are 32-bit numbers written in four dotted decimal octets (192.0.2.1); IPv6 addresses are 128-bit numbers written in eight hexadecimal groups. Every packet on the internet has a source IP and a destination IP, and routers use those addresses to forward the packet toward its destination. Without an IP, a device is invisible on the network.
What an IP Reveals
An IP address is mapped, with reasonable accuracy, to a geographic region — typically a city or metropolitan area, though sometimes only a country. It is associated with an ISP or hosting provider, which can be looked up via WHOIS. It is the primary key in many tracking and abuse databases. Combined with timing, it can be linked back to a specific subscriber by their ISP. For these reasons, hiding the IP is the headline feature of every consumer VPN.
VPN and IP
When you connect through a VPN, the destination sees the VPN server's IP, not yours. Your ISP sees your traffic going to the VPN server's IP, but cannot see beyond it into the encrypted tunnel. The IP shown to the destination is shared with however many other users are on the same VPN server (in a shared-IP model) or unique to you (with a dedicated IP). Either way, your home IP is not exposed.