Many modems have a router built-in. If this is the case and you add an Airport Extreme or Express to it you don't want that to be handing out numbers as well as the router - that's a recipe for chaos.
Fortunately if you have Lion or above and an up-to-date Airport you don't have to concern yourself about this.
MEET THE WIZARD
When you initially set up your Airport to add it to your modem or modem/router the 'wizard' will automatically detect how it is connected, suggest the correct type of setup, and prompt you to create a name for the Airport, and a name and password for the wireless network you are creating or extending. Open Applications/Utilities/Airport Utility; select your Airport (or Time Capsule - effectively the same thing) and follow the instructions to create a new network or extend an existing one. Page 21 of the manual outlines the procedure, and Bob Timmons has provided an illustrated walk-through in the Apple Support Forums (he is answering a question about the Time Capsule but it's the same for an Airport).
However, the wizard will only function on first set-up. If you are amending an existing arrangement and don't want to completely reset your Airport, or if you are using a pre-Lion system (or if you just want to do it yourself) it's not difficult to set up manually.
Your Airport has three possible modes:
1. 'Bridge mode' is used when the AE is connected to a modem which is also a router, and so which distributes IP addresses. The modem/router allocates an IP number to each device and the AE simply passes it through. (You don't want two devices on the same network handing out IP numbers - if you want the AE to hand out the numbers you have to be able to set the router part of the modem into bridge mode, which may not be possible in many cases).
2. 'Distribute a range of IP addresses' is used only when your ISP provides you with a number of IP addresses, so that one of these can be allocated to each device. Very few ISPs, if any, do this.
3. 'Share a public IP address' is used when your ISP allocates only one IP number (the usual case) and your modem does not have a router. The AE the handles the distribution of local IP numbers to the various devices, receiving its own single IP number from the modem.
When your ISP allocates only one IP number, using either modes 1 or 2 results in that single number being allocated to the first device which connects. Connecting a second device gets that same IP number allocated to the second device, so the first device loses its number and the connection along with it.
Assuming that your modem also includes a router, the router part issues your various devices with new IP numbers - Network Address Translation - which are in a range of numbers for local use only. These numbers cannot be seen by the outside world.
If you now connect an Airport Extreme or Express and it starts issuing its own IP numbers, there is a danger that it will decide to issue a number which the first router has already put into use - 'double NAT'. This results in two devices with the same IP number, and chaos. In some cases you can get away with this as the AE probably issues numbers beginning with 10, and the other router may issue numbers beginning with 192.
However it's better to put the AE into 'bridge mode', so that it simply passes through the IP numbers allocated by the first router. This works perfectly well - I've done it myself for years and never had any problems at all.
The exact method varies with your version of the Airport Utility. On older versions, open Applications/Utilities/Airport Utility. Select your Airport in the list and click 'Manual setup'. Click 'Internet' in the toolbar. Click the 'Internet Connection' tab if necessary. Set the 'Connection Sharing' menu to 'Off (Bridge Mode)'. Click 'Update' and wait for your AE to restart.
On the current version, open Applications/Utilities/Airport Utility. Click the image of your Airport and in the pop-up pane click ‘Edit’. In the pane which drops down, click the ‘Network’ tab. Set ‘Router Mode’ to ‘Off (Bridge Mode)’ and click ‘Update’; wait for your AE to restart.
Having done this you can now connect devices to the other ethernet ports on your modem/router, the ethernet ports on your Airport, or the wireless networks created by either the modem/router (if it has wireless) or the Airport and all will connect simultaneously to the internet; and any devices which allow file sharing can connect to any other devices which allow it.
Roger Wilmut.This site is not associated with Apple.