The value after the slash, i.e. the 24 in your example 192.168.1.0/24, uses CIDR notation to
indicate the number of bits available for network addressing as distinct from host addressing. For IPv4 each IP address is 32 bits, and the network address for a /24 network has 24 bits, so the host addressing for such a network would be 32 - 24 = 8 bits.
Let's look at this a bit more closely.
Take an example address 192.168.1.0/24. This says that 24 bits of the 32 are for the network address. Each octet is 8 bits so it become trivially easy to see that this means that 192.168.1 is the network address, and the remainder is for the host. Eight bits gives 28 addresses, i.e. 256. The lowest (0) is unavailable and the highest (255) is reserved for local network broadcasts, so that leaves room for 254 host addresses, all beginning with 192.168.1.
Now take another example address 192.168.0.0/16. Here we have 16 bits of the 32 for network addressing, leaving 16 bits for the hosts on the network. We have 216 = 65536 host addresses but, as before, two are reserved (192.168.0.0 and 192.168.255.255) so you have 65534 available addresses for hosts on this network, all starting 192.168.
This is all very easy; where it gets exciting is when the subnet field is not a multiple of eight. For example, you could have a network 192.168.1.128/26. The same rules apply though; you have 26 bits for the network address and 6 bits for the hosts on that network. 26 is 64 and two are reserved so you can have 62 hosts on such a network. Using the ipcalc tool you can see that the valid IP addresses on this network would be 192.168.1.129 to 192.168.1.190:
ipcalc 192.168.1.128/26
Address: 192.168.1.128 11000000.10101000.00000001.10 000000
Netmask: 255.255.255.192 = 26 11111111.11111111.11111111.11 000000
Wildcard: 0.0.0.63 00000000.00000000.00000000.00 111111
=>
Network: 192.168.1.128/26 11000000.10101000.00000001.10 000000
HostMin: 192.168.1.129 11000000.10101000.00000001.10 000001
HostMax: 192.168.1.190 11000000.10101000.00000001.10 111110
Broadcast: 192.168.1.191 11000000.10101000.00000001.10 111111
Hosts/Net: 62 Class C, Private Internet