The thing that assigns you an ip address is the dhcp server (not client).
The DHCP client is a small discovery program which will go out and find an available dhcp server. If there is a server available then the client will ask the server for network information. The server will then provide the ip address, the network mask, the gateway address, and the dns servers. The client will then use this information the dhcp server provided information to access the network.
There is also lease time on the client ip addresses. Once a certain amount of time passes (1 hour, 10 days, it could be set to anything), the client will need to renew it’s settings. It basically starts the whole discovery process again but if the dhcp server can give the dhcp client the same ip address, it will.