A company called Network Wizards, does a biannual DNS walk with a robot called ZONE. Essentially, this means they have counted the number of hosts on the Internet. If you are unclear on the distinctions between hosts, web sites, IP addresses and individual computers, see the terminology page.
The number of hosts on the Internet has roughly tripled in the time from January 1994 to January 1996. In order to understand the numbers more specifically, or to quantify exactly how many machines are currently 'on the Internet', it is necassary to define precisely what one means by 'on the Internet'. There are a few possible definitions:
On the Internet means that there is a listing for a hostname and IP address on a name server. The machine does not actually have to exist, but by virtue of its mention on a name server, it is counted.
On the Internet means that the machine is both listed on a name server and is responding to 'ping', an Internet utility to determine if a machine is on and connected to the network. This will exclude machines that are behind restrictive firewalls (a way of increasing network security) or machines that are not on at the time of the pinging.
On the Internet means some fuzzy definition in between these two previous definitions.
The last definition is probably the best, but is the hardest to measure. The Network Wizards survey provides numbers given the first two definitions.
1.7 million hosts on the Internet in January 1996, up from
1.0 million in 1995,
0.6 million in 1994, and
less than 0.4 million in 1993,
by the second definition above
The raw data is available.