This article is from the Pool & Billiards FAQ, by Bob Jewett with numerous contributions by others.
There are many diamond systems. In general, they allow you to plan
shots that require the cue ball or object ball to contact one or more
rails. One of the best treatments for pool is in Eddie Robin's first
one pocket book. A large part of Byrne's books and articles are about
diamond systems. Walt Harris has four books out ("Billiard Atlas
[1-4]") that cover mostly carom diamond systems, but he also discusses
their use on pool tables.
If a system is called simply the "Diamond System" the speaker probably
means the "corner five" system. A shot from that (for pool tables):
Place the cue ball as shown, and shoot it to "a" with running english
(side spin). The cue ball should hit cushions at a-b-c and go towards
the other corner to pocket "o". The system tells you how to adjust to
go to any destination on the third rail from any origin for roughly the
same kind of path. This is done by assigning numbers to the spots
(diamonds) on the rail and doing some simple arithmetic. See Byrne's
"Standard" book for use of the formula. Tables, balls, stroke and
sticks vary. A very simple example:
___________ ____________ q c | | (Use a fixed-width font to view this.) | | ("Courier" might work.) | b| | | "a" is roughly 2 diamonds from the corner | | "c" is roughly 3 diamonds from the corner o__________ ______a_____ "q" is at the corner which is assigned "5"