The magic word is DISPLAY. In the X window system, a
display consists (simplified) of a keyboard, a mouse and a
screen. A display is managed by a server program, known as an X
server. The server serves displaying capabilities to other
programs that connect to it.
A display is indicated with a name, for instance:
DISPLAY=light.uni.verse:0
DISPLAY=localhost:4
DISPLAY=:0
The display consists of a hostname (such as
light.uni.verse and localhost), a colon
(:), and a sequence number (such as 0
and 4). The hostname of the display is the name of
the computer where the X server runs. An omitted hostname means
the local host. The sequence number is usually 0 -- it can be
varied if there are multiple displays connected to one computer.
If you ever come across a display indication with an extra
.n attached to it, that's the screen number. A
display can actually have multiple screens. Usually there's only
one screen though, with number n=0, so that's the
default.
Other forms of DISPLAY exist, but the above will do
for our purposes.
For the technically curious:
hostname:D.S means screen S on
display D of host hostname; the X
server for this display is listening at TCP port
6000+D.
host/unix:D.S means screen S on
display D of host host; the X server
for this display is listening at UNIX domain socket
/tmp/.X11-unix/XD (so it's only reachable from
host).
:D.S is equivalent to host/unix:D.S,
where host is the local hostname.