I have two IP addresses. They share an IP address to the nth byte. Is there a word that effectively says that they have the same IP address up to the nth byte? Are there different words depending on how many successive bytes they have in common?
First, an IP address is specified in bits, not bytes. Since humans tend to have a difficult time with binary, it is often converted into dotted decimal or bytes. This is an important distinction.
For example, two hosts share the same network of 10.10.10.0/23, they could have IP addresses of 10.10.10.3 and 10.10.11.3. They share the same 23 bits, but they only share the first two bytes, however the first two bytes will also include hosts that are not part of the network such as 10.10.255.3.
As to what the shared bits are called, this can be a number of things. Typically it will be network or net or subnetwork or subnet with or without ID or prefix appended (for example network ID or subnet prefix).
I am sure I have probably seen it called other things as well, but those combinations seem to provide everything I can think of at the moment.
The word you are looking for is probably network prefix or subnet mask. It is used in conjuntion with an address mask.
192.168.0.0 / 24 or 192.168.0.0 / 255.255.255.0
"24" in means that the 24 most significant bits (3 bytes) of the ip address are the network prefix, the rest is available for addressing (without the .0 which is the subnet address and 192.168.0.255 which is a broadcast address for the network)
More on that topic on wikipedia: http://en.wikipedia.org/wiki/Subnetwork