How to understand the "A subnet Example"?

Routers can be any IP address on the network, though they are usually the first number or the last number, but this is only by convention.

The subnet ID in this example is where the subnet range starts.

For example, take the subnet 192.168.1.1/24. A /24 subnet has 255 IP addresses in it. .255 is the broadcast IP and 1-254 can be used by any node on the subnet-- servers, routers, etc. The subnet ID just states that this is where the subnet range starts.

Here's another example using a less standard subnet. A subnet with a CIDR prefix of /26, or a subnet mask of 255.255.255.240. For this example, it will be 192.168.1.1/26. This takes the 192.168.1.1/24 subnet and splits it up into into 4, subnets, each with a size of 64.

Thus, 192.168.1.1/26 splits 192.168.1.1/24 into subnets with IDs of:

192.168.1.1, 192.168.1.64, 192.168.1.128, 192.168.1.192

192.168.1.1/26 gives us the subnet ID (which can also be used for a node on the subnet). The last address in this subnet will be 192.168.1.63, which will be the broadcast address for this subnet.

the same goes for the other 3 networks created:

192.168.1.64/26 with a broadcast address of 192.168.1.127

192.168.1.128/26 with a broadcast address of 192.168.1.191

192.168.1.192/26 with a broadcast address of 192.168.1.255

It should also be noted that in many home networks, the DHCP server is shared by the router, so these will often be the same IP address. In most corporate networks, DHCP servers can be on any IP address, they do not need to be shared with the router. And as I said, the router can technically be on any address, but by convention the router is usually the first or last address in the subnet.

Here is my favorite subnet explanation chart:

These are net ID's, which the first network adresses are called. These subnet prefixes have /27 mask (255.255.255.224 in dot notation), thus they are starting from .32, .64, ./96 (and so on) octets. First IP in the network isn't useable in IPv4. So is the last one.