When I change the subnet for the LAN interface on a Sonicwall firewall the WAN interfaces go haywire. How to do

Many Answers here have given textbook-like definitions that have touched on this, but there are certain practical, real-world distinctions worth underscoring:

A LAN is typically something completely within your own premises (your organization's campus, or building, or office suite, or your home), so it's something you build and own yourself, all the way down to the physical cabling. In contrast, a WAN is something that connects between geographically separated locations, so you generally have to lease access to lines or data transmission services from telecommunications carriers to create your WAN.

Because of the shorter distances needed for on-premises networks, and because of the focus on connecting PCs and servers, LANs tend to be built on Ethernet and other 802.3-family (and 802.11-family) physical layers and data links. 1000BASE-T, 1000BASE-SX, 1000BASE-LX, etc.

Because off-premises network links usually need to go longer distances and work over the telecommunication carriers' existing infrastructure, they tend to use physical and data-link standards that are more common in the telecom industry. Then again, for your convenience, telecom carriers usually hand off to you using an 802.3-family link, even if what they're using behind the scenes is, say, OC-3 and SONET/SDH.

Because of the technical hurdles associated with moving lots of data long distances reliably, WAN links tend to be lower bandwidth and higher latency than LAN links. Also, because you're usually paying a separate telecom carrier for the service, to keep costs down, most organizations try to limit (or at least not go crazy with) how much data they move over WAN links.

The differences between LANs and WANs tend to be at the physical and data-link layers. At the network layer (Layer 3 in the old-but-still-helpful-for-some-discussions OSI layering model), most people use IP (Internet Protocol) nowadays. Because it all uses IP, applications that use IP don't have to know what physical and data-link layers are in use, so anything you can do on a LAN you can potentially do on a WAN as well, if you have high enough bandwidth and low enough latency for whatever it is you're trying to do, and as long as you haven't blocked it at your firewall (or via the accidentally firewall-like properties of a NAT).

The generic name for both methods is Cascading, although the second one is sometimes more accurately called bridging. The router connected to the Internet is called the main router, while the other one is called the secondary router. Cascading or bridging can be used to extend the network's range and/or to reduce the number of devices communicating with each router. It cannot increase the total Internet bandwidth of the network beyond the capacity of the main router.

LAN to LAN

Connecting one of the Ethernet ports (or LAN ports) of the main router to one of the Ethernet ports of the secondary router.

This type of cascading forms a bridge between both routers and both wireless networks, and requires the main and the secondary routers to be on the same LAN IP segment to allow the computers and other devices to connect to both routers. To do this, you need to disable the secondary router's DHCP server, so that router is really degraded to a simple switch in bridge mode. This configuration is recommended if you want to share files and resources within the network.

The advantage of this setup is that all devices are effectively on the same LAN (i.e. bridged) and can communicate with any protocol without additional setup. It is also compatible with practically any router on which you can turn off DHCP, as the router does not have to perform any layer-3 (IP) work at all.

Another key advantage is because it is on a single bridged network, if you set both routers to the same SSID and security on both routers, your devices can seamlessly roam between both routers, connecting to whichever has the strongest signal and not disconnecting you when moving between them.

The downside of this, is the same as you get with any large bridged network - increased broadcast traffic, which can impact mobile device battery life.

LAN to WAN

Connecting one of the Ethernet/LAN ports of the main router to the Internet port (WAN port) of the secondary router.

This type of cascading requires the main router and the secondary router to have different IP segments. This connection makes it easier to identify which router the computers and other devices in the network are connected to since they will have different LAN IP segments. However, computers that are connected to the main router will not be able to communicate with the secondary router without additional configuration, and vice versa since there are two different networks.

In general, this is a less preferred method, as it requires additional configuration (manual/static routing) that isn't always possible on consumer routers. Further, again for consumer routers, it gives you a double-NAT situation for devices behind the second router, which is undesirable. The additional layer-3 NAT/routing work imposed on the routers' CPU may also decrease wireless speeds

A final drawback is that the separate subnetting means you cannot move between the two networks automatically - a device must fully disconnect from one network and connect to the second one, it will not automatically switch to whichever router has the strongest signal.