What's the meaning of 10.0.0.1/24 address of my computer (from the "ip addr" command)?

Thought I would expand on this with a few examples

/8 = 255.0.0.0

/16 = 255.255.0.0

/24 = 255.255.255.0

/32 = 255.255.255.255

192.168.1.0/24 = 192.168.1.0-192.168.1.255

192.168.1.5/24 is still in the same network as above we would have to go to 192.168.2.0 to be on a different network.

192.168.1.1/16 = 192.168.1.0-192.168.255.255

When you have a network you lose two IP addresses one for broadcast and one for the network. The first IP is reserved to refer to the network while the last ip of the range is reserved for the broadcast address.

RFC 1918 reserves 3 ranges for private IP addresses. Your DHCP server/router is configured to assign this range.

10.0.0.0 - 10.255.255.255/8

172.16.0.0 - 172.31.255.255/12

192.168.0.0 - 192.168.255.255/16

http://en.wikipedia.org/wiki/Private_network

In addition to Tim's answer:

The /24 instead of /8 means that the first 3 octets of the ip address are used to specify the network. This is just a setting you can change if you want to. It's not super common to use the 10. private range with a /24 mask but there's no reason you can't do it.

/8 is using only the first octet to specify the network portion, which is what a 10. network explicitly meant back in the pre-CIDR days, and that's why you still see it more often with a /8 than with a 24.

As for the last octet being a 0 not a 1, that's because a 10.0.0.0 would in this case be the network address, with 10.0.0.1 being your computers ip.