This article is from the Go FAQ, by Morten PAHLE firstname.lastname@example.org with numerous contributions by others.
The usual convention is to label the Goban from left to right with
letters from A to T (omitting 'I' to avoid confusion with '1') and
from bottom to top (as seen by black) with numbers from 1 to 19.
In Japan, the coordinates are all numbers; arabic (1,2,3) along one
axis, and Kanji along the other.
A move is then referred to by colour and position, e.g. Black E4.
When publishing moves (in e.g. a book or an article, or in a post to
the newsgroup - see also section 4.6) the most often used method is to
represent (parts of) the Goban with numbers at the intersections.
In publication-quality diagrams, black and white circles have move
numbers inside them. Alternative moves or stones/moves discussed in
the text are labelled with letters or symbols (triangles, etc.). Even
publication-quality diagrams can get very crowded in a typical game
(try finding move 131 in a diagram containing over 200 moves), so
often they are broken up into multiple diagrams of 50 moves each, or
fewer (the fewer moves per diagram, the easier the game is to read).
To record games by hand, there are pre-printed sheets available with a
19x19 grid on them. As play goes on, you label each intersection with
the move number. Some people use two pens with different colours (e.g.
red and blue) for this, distinguishing black stones from white. Others
put circles around the black moves, but not around the white moves, or
similar. Notes such as "201 at 47" are made at the bottom for kos and
other plays "under the stones" (playing on an intersection after the
stone originally placed there is captured). This does become
cumbersome after a while, though.
If you play on a computer, your program (player, or client) records
the game for you. Games on the Go servers (see section 3.1.) are
stored as SGF files; some programs use other formats as well (see
section 6.2). Using a server's "automail" function, you can have all
your games automatically e-mailed to you, for easy review with SGF
Programs which allow you to view, edit and comment these files exist
for almost all operating systems. See section 6.1.