https://en.wikipedia.org/wiki/Responsibility_assignment_matrix
RACI letter meanings:
R = Responsibility ("the doer")
A = Accountable ("the Buck stops here")
C = Consulted ("in the loop")
I = Informed ("kept in the picture")
To build a responsibility assignment matrix:
Build a matrix with a list of tasks/activities to be performed in one dimension (typically the vertical one) and a list of roles (or even resource names) in the other dimension, then think who needs to be R, A, C and I. Keep in mind that 2 Rs in the same row might create ambiguity and probably are not desirable to be kept like that (the effect of: "I thought the other party/person would do it!").
Several variants exist, and as for the SMART acronym (used for tasks, requirements, ...) sometimes a letter means different things between each of the variants. So read on (quoting Wikipedia):
3 RACI
3.1 PARIS
3.2 PACSI
3.3 RASCI
3.4 RASI
3.5 RACIQ
3.6 RACI-VS
3.7 CAIRO
3.8 DACI
3.9 RAPID
3.10 RATSI
3.11 DRASCI
4 Variations
4.1 RACI (alternative scheme)
4.2 ARCI (decisions)
One example:
https://commons.wikimedia.org/w/index.php?curid=26348185 (source: Rickysmithcmrp - Own work, CC BY-SA 3.0)
Also bear in mind that there is a distinction between a role and individually identified people:
Quoting:
"A role is a descriptor of an associated set of tasks; may be performed by many people; and one person can perform many roles. For example, an organization may have ten people who can perform the role of project manager, although traditionally each project only has one project manager at any one time; and a person who is able to perform the role of project manager may also be able to perform the role of business analyst and tester."
Updates: 2018-02 - role vs people
