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.