Dynamic Host Configuration Protocol
In simple terms, DHCP determines if your IP is static or dynamic and the length of time an IP address is assigned.
Dynamic Host Configuration Protocol is controlled by a DHCP server. Your router can be a DHCP server...and if you're on a home network, it most likely serves this purpose.
I know this might be confusing because the word dynamic is in the term, but just because you have DHCP enabled on your computer doesn't mean you can't be assigned a static IP. DHCP enabled on your computer simply means you're letting a DHCP server assign its IP. Having it enabled DOES NOT mean it's a DHCP server.
A true DHCP server (not your Linksys router) gives the LAN Admin a ton of control with IP assigning.
Ever print to a network printer? Ever wonder how that printer keeps its network assignment? Each network device has a MAC address. You can assign a static IP at the server to a specific MAC address. This allows the network printer to always get the same IP even after it reboots and without assigning the IP at the printer. If you print the network configuration at the printer, it will probably tell you that DHCP is enabled and no static IP is assigned. That's because the IP assignment is handled at the server.
Your ISP has a DHCP server. They can assign IPs by modem MAC addresses. When your modem comes online, it communicates to the network indicating it is looking for an IP address. The DHCP server listens to this communication and starts talking to the modem. The modem then transmits its MAC address to the DHCP server. At that point, either an IP has been reserved for the modem or one is assigned at that time. Hence cloning your MAC address to get a new IP from your ISP.
Routers and DHCP - Under the General Setup or LAN Setup tab in your router, you'll see a settings option for DHCP. You can control how many IPs are assigned or to enable/disable the DHCP server portion of the router. If you disable it, you'll have to statically assign IPs to each computer, or have a DHCP server or your network. This goes for wired and wireless. Any connection on your network has an IP address.
There's a lot more to DHCP than this, but this is a basic explanation.