Each time a DHCP client boots, it sends out a DHCP discover message. All DHCP servers answer (in practice only one is set to do this at Fermilab; in the future possibly a second will be added for redundancy) with an offer message that includes an address which is available to the client.
The client machine typically repeats the discover message several times to make sure it hears from all the servers, then eventually chooses the "best" server, where what is "best" is up to the client. It may mean that the addresses the DHCP server has available offer the longest lease time. Or the client might prefer a server that provides WINS servers over one that doesn't (the WINS servers keep track of all the clients' and servers' latest dynamic IP addresses).
The currently active DHCP server is configured by hand to handle and reserve IP addresses and the IP configuration information that goes with them. Addresses are made available in an order that permits a client to have the best chance of getting back the same address it was using most recently. To this end, the DHCP server offers its least recently used address to a new client.
Once the client chooses a DHCP server, it "officially" requests the IP address and configuration information. In addition to this, it receives a lease time for the address. This lease time is not absolute. As long as it is running, the client machine requests renewal of the lease. This is invisible to the user, although there is a mechanism for the user to release the address early ( ipconfig/release from the command prompt).
Client machines in the NT domain typically access multiple file servers, print servers, and so on. The clients as well as the servers may change their IP addresses. Via the WINS servers, this is transparent to the user.