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I will be using the official WCA notation from Article 12 of the WCA Regulations:

Article 12: Notation

12a) Notation for Rubik's Cube and similar puzzles:
Face Moves:
  • 12a1) Clockwise, 90 degrees: F (front face), B (back face), R (right face), L (left face), U (upper face), D (bottom face).
  • 12a2) Counter clockwise, 90 degrees: F', B', R', L', U', D' (see 12a1).
  • 12a3) Clockwise, 180 degrees: F2, B2, R2, L2, U2, D2 (see 12a1).
  • 12a4) Counter clockwise, 180 degrees: F2', B2', R2', L2', U2', D2' (see 12a1).
Double Outer Slice Moves (outer slice plus adjacent inner slice):
  • 12a5) Clockwise, 90 degrees: Fw, Bw, Rw, Lw, Uw, Dw. (see 12a1).
  • 12a6) Counter clockwise, 90 degrees: Fw', Bw', Rw', Lw', Uw', Dw' (see 12a5).
  • 12a7) Clockwise, 180 degrees: Fw2, Bw2, Rw2, Lw2, Uw2, Dw2 (see 12a5).
  • 12a8) Counter clockwise, 180 degrees: Fw2', Bw2', Rw2', Lw2', Uw2', Dw2' (see 12a5).
Middle Slice Moves (middle slice of puzzles with odd number of slices, middle two slices of puzzles with even number of slices):
  • 12a13) Clockwise, 90 degrees: M (same direction as L), S (same direction as F), E (same direction as D). (see 12a1).
  • 12a14) Counter clockwise, 90 degrees: M', S', E' (see 12a13).
  • 12a15) Clockwise, 180 degrees: M2, S2, E2 (see 12a13).
  • 12a16) Counter clockwise, 180 degrees: M2', S2', E2' (see 12a13).
12b) Rotations for all cube shaped puzzles:
  • 12b1) Clockwise, 90 degrees: [f] or z, [b] or z', [r] or x, [l] or x', [u] or y, [d] or y'. (see 12a1).
  • 12b2) Counter clockwise, 90 degrees: [f'] or z', [b'] or z, [r'] or x', [l'] or x, [u'] or y', [d'] or y (see 12b1).
  • 12b3) Clockwise, 180 degrees: [f2] or z2, [b2] or z2, [r2] or x2, [l2] or x2, [u2] or y2, [d2] or y2 (see 12b1).
  • 12b4) Counter clockwise, 180 degrees: [f2'] or z2', [b2'] or z2', [r2'] or x2', [l2'] or x2', [u2'] or y2', [d2'] or y2' (see 12b1).

To make things easier to explain, it is useful to develop a notation that easily describes certain moves.  For the purposes of the diagrams below, F is the red face, B is the orange face, L is the blue face, R is the green face, U is the yellow face, and D is the white face:

Standard Face Turns
Thus, I will use this notation in which each face is represented by a letter (F (front), B (back), L (left), R (right), U (up), and D (down)) and moves can be described by the letter of the face that is being turned.  An upper-case letter means to turn that face in the clockwise direction.  If that letter is followed by a prime ('), then that face should instead be turned counter-clockwise.  Similarly, if that letter is followed by the number two (2), then that face should be turned twice.  Sometimes, I will follow a prime with a 2 ('2), indicating that the move should be made in the counter-clockwise direction (for speed reasons).  A capital letter followed by a lower case "w" works the same way, except instead of just turning the outside face, the middle layer adjacent to it should be moved as well (in the same direction).  For example, "Dw" is equivalent to performing "D" and "E" simultaneously.

F F2 or F'2 F' Fw Fw2 or Fw'2 Fw'
B B2 or B'2 B' Bw Bw2 or Bw'2 Bw'
L L2 or L'2 L' Lw Lw2 or Lw'2 Lw'
R R2 or R'2 R' Rw Rw2 or Rw'2 Rw'
U U2 or U'2 U' Uw Uw2 or Uw'2 Uw'
D D2 or D'2 D' Dw Dw2 or Dw'2 Dw'

Slice Moves
Slice turns (M, E, and S) are sometimes useful for certain algorithms.  In a slice move, the "outside" layers remain stable, and a middle layer will slice through them.  They can have the same suffixes as the other face turns.

M M2 or M'2 M'
E E2 or E'2 E'
S S2 or S'2 S'

Cube Rotations
Cube rotations (x, y, and z) do not involve the turning of any layers of the cube.  Instead, the entire cube is rotated (imagine an x, y, and z axis going through the cube).  These are often used to move the cube into a position that makes an algorithm easier to perform.

x x2 or x'2 x'
y y2 or y'2 y'
z z2 or z'2 z'