Do I still need my Apple AirPort Extreme?

I assume this means I can eliminate the AirPort Extreme and set up my network using only the router (?)

Correct.

Or does Airport serve to extend the range of the wifi connection to the entire house?

No.....unless.....you locate the AirPort Extreme at the other end of the house and connect it back to your Arris gateway using a permanent wired Ethernet cable connection. If you want to run the wiring, post back for more instructions on how to configure the AirPort Extreme for this type of setup.

So if I understand you correctly, the AirPort Extreme (and the Airport Express that pre-dated it) were only necessary because, at the time I only had a simple modem and needed it to create a wifi network.

Correct. A simple modem has only one Ethernet port, so you can only connect one device to it. That is not a problem if you only have one computer. But....if you want to have wireless network or even connect more devices using a wired connection, you need to have a router, which connects to the modem and then your network devices connect to the router.

Some time ago, most manufacturers began to offer both a modem and a router in one single package......because it made things much simpler for the vast majority of users. Who did not want to fool with setting up two devices, both a modem and router.

So, modem/routers, also known as gateways, have become very popular. Unfortunately, most users still call a modem/router or gateway a "modem"......even though it is also a router.

Apple could have made things a lot easier if they had decided to sell modem/routers or gateways, but they never did, except for one ancient device. The unfortunately truth is that most folks who bought an AirPort router did not really need it......because they also had a modem/router or gateway.

It did not hurt to have the AirPort with a modem/router, but it also did not really help.

can you tell me (in terms I can understand), how to set up the new network using only the router?

It should already be set up. Look for the name of the wireless network that the modem/router is providing and connect your devices to that network. For example, if the modem/router provides a wireless network signal with a name like......."Arris Wireless".......you would connect your devices to that network.

Information about the wireless signal and password is sometimes located on a label on the back or bottom of the product.

Sorry, I can't help on the specifics of the modem/router or gateway that you have. If your Internet Service Provider furnished the product to you, you need to get with their support folks for additional information.

If your ISP has provided you with a modem/router or gateway type of device which combines the functions of a separate modem and router in one single package, then you really don't need the AirPort Extreme.

Here I am assuming that the ISP modem/router or gateway provides a Wi-Fi signal as well, correct?

It would really help clarify things if you could provide the make and model number of the device that you likely call your "modem".........but chances are, it is likely more than a simple modem.

A simple modem looks like the example below. It has only one Ethernet port so it is not a router, nor can it provide a Wi-Fi signal:

An example of a modem/router, which is also called a gateway is shown below. Note that the device will have multiple Ethernet ports and produce a Wi-Fi signal.

The bottom line.......

If your provider has furnished you with only a simple modem. then you will need a router like an AirPort Extreme or another brand of router.

If your provider has furnished you with a modem/router or gateway, then you don't need another router like an AirPort Extreme or any other router for that matter.

There really is no such thing as an "Apple" network. A network (wired and wireless) can be provided by any networking hardware manufacturer, which Apple is not longer participating as.

For example, I completely transitioned from an all-Apple HW network to that provided by Ubiquiti. As a result of doing so, it had no impact (other than better performance) with any of my Apple devices ... that includes Apple TVs, iPhones, etc.