Okay, here's how DHCP works.
Dynamic Host Configuration Protocol is a method of automatically assigning IP addresses. Basically, the DHCP server sets up a 'pool' of available IPs. Those IPs are assigned to various clients, be they computers, printers, or something else, automatically, on a first-come-first-served basis. Each of those IPs is 'leased' for a certain time period, typically 8 days. (The lease can be much shorter or much longer, but everyone gets the same lease) The server notes the MAC address of the device to which that lease was made; if, during the period of the lease, that device drops off the network, the IP that was leased to it is
not released to another device. If the device comes back up before the lease is over, it gets that IP back. If it comes back up on the net
after the lease is over, it gets another IP because when the lease is over that IP is released for use elsewhere, and on a busy net will have been grabbed already. If the device is using the IP when the lease expires, the lease will be renewed. When I was with Comcast, their system set leases to expire every 4 days, but I had the same IP for nearly 2 years, because I very rarely turned my cable modem off, and the lease autorenewed every time.
The typical home network uses a Class 'C' private network range, usually 192.168.0.x or 192.168.1.x, where 'x' is a number between 1 and 254. (You can't use a '0' or a '255' with a Class 'C' for technical reasons.) Most home routers come set to use a chunk of the Class 'C' range, typically from 101 up, as the DHCP pool. They'll use 1 as the address for the router itself, and IPs from 2 to 100 can be used as fixed IPs. IPs from 101 up would be assigned by DHCP. If your router is like that, you can use your printer admin tools to set the printer's IP to something like 192.168.0.10.
If you can't find the printer admin tools but can access the router's control page (typically found at 192.168.0.1, the router's address; see your router's Read Me for the address of your particular router) using any web browser, you can tell the router to assign certain IPs to specific devices. What you need is to get the MAC address of the device you want to assign. In the case of the HL2070N, you need to awaken the printer (touch the 'Go' button once) and then, once the green LED shows that the printer is awake, touch the 'Go' button three times in rapid succession. That will print the Print Setting pages, three pages with the current settings of your printer. The MAC address is on page 3, under 'Network Configuration'; Brother calls it the 'Ethernet address'. Write that MAC address down, go to the router control page, and reserve an IP for that address. The router will
always hand that IP to the printer. If the printer is not available, no other device will ever get that IP. You get the benefit of a fixed IP without having to play with the printer's controls. This is called 'reserving an IP', and almost all modern home routers can do it.
If your router knows about Bonjour you don't even have to do that, but not all routers use the Bonjour protocols. If Bonjour is live, you merely turn your router, printer, and Mac on, give the router a minute or two to stabalise and hand out IPs, and you can print.
