Okay, the Airport Express is what is commonly called a "router" or a "gateway." Its primary purpose is to provide internet access to computers by sharing a "wide" area network (WAN) with a "local" area network (LAN).
The usual configuration is that you plug it into power, plug an ethernet cable from a device with an internet connection (a cable modem, a DSL modem, etc -- this is the WAN) and then it provides this connection wirelessly to your computers (in the LAN).
You cannot do that unless you have an internet source plugged into the Airport Express physically (ethernet cable).
Routers can also allow computers within the local area network to share files and so on (actually, with Macs, you don't need a router for this, but that's a side issue...).
In addition, the Airport Express has a couple of special functions most routers don't have. If you plug an audio cable between it and your stereo, you can play songs from your computer wirelessly to the Airport Express, and from there by wire to the stereo and its speakers. Finally, if you plug a printer into the Airport Express by USB, you can print to it wirelessly from the computer to the Airport Express (and then by wire from the AirEx to the printer).
These features in the last two paragraphs do NOT require a WAN connection. So in principle, you could use them in the type of situation you describe, although probably that wouldn't make any sense....