DHCP is short for Dynamic Host Configuration Protocol, and is a protocol used to assign IP addresses to devices in a network.
DHCP is managed by a DHCP server. In a home network, this probably means the network router which assigns dynamic ip addresses to the connected computers.
A real DHCP server allows for a lot more options though, for example assigning an IP address to a specific MAC address. This is useful if you have for example a printer that needs to keep the same IP address, since it has been configured as a network printer on other devices in the network.
ISP's uses a DHCP server to assign external IP's to their customers. This IP can be static or dynamic, determining if you change IP or stick to the same one all the time.