Notation note: A letter (e.g. R) means turn that face 90° clockwise as you look directly at that face. An apostrophe (') means 90° counterclockwise (inverse). 2 means a 180° turn.
R - Right face clockwise (90° clockwise when looking at the Right face).
R' - Right face counterclockwise (inverse of R).
R2 - Right face 180° turn.
L - Left face clockwise (90° clockwise when looking at the Left face).
L' - Left face counterclockwise.
L2 - Left face 180° turn.
U - Up (top) face clockwise (90° clockwise when looking at the Up face).
U' - Up face counterclockwise.
U2 - Up face 180° turn.
D - Down (bottom) face clockwise (90° clockwise when looking at the Down face).
D' - Down face counterclockwise.
D2 - Down face 180° turn.
F - Front face clockwise (90° clockwise when looking at the Front face).
F' - Front face counterclockwise.
F2 - Front face 180° turn.
B - Back face clockwise (90° clockwise when looking at the Back face).
B' - Back face counterclockwise.
B2 - Back face 180° turn.
Rw (or r) - Right wide: turn the two rightmost layers together (the R face plus the adjacent middle slice). Equivalent to R + the middle slice turn in the same direction.
Lw (or l) - Left wide: turn the two leftmost layers together.
Uw (or u) - Up wide: turn the two topmost layers together.
Dw (or d) - Down wide: turn the two bottom layers together.
Fw (or f) - Front wide: turn the two front layers together.
Bw (or b) - Back wide: turn the two back layers together.
M - Middle slice between L and R. Turns like L (i.e. same direction as L face turn).
M' - Middle slice inverse.
M2 - Middle slice 180° turn.
E - Equatorial slice between U and D. Turns like D (i.e. same direction as D face turn).
E' - Equatorial slice inverse.
E2 - Equatorial slice 180° turn.
S - Standing slice between F and B. Turns like F (i.e. same direction as the Front face turn).
S' - Standing slice inverse.
S2 - Standing slice 180° turn.
x - Rotate the whole cube like an R move (rotate cube on the x-axis: R). x' and x2 as usual.
y - Rotate the whole cube like a U move (rotate cube on the y-axis: U).
z - Rotate the whole cube like an F move (rotate cube on the z-axis: F).
Common shorthand: Lowercase letters (r, l, u, d, f, b) are often used to mean wide turns (two layers) in some literatures, but other sources prefer Rw, Lw... - both conventions are common
Tip: When in doubt about direction, imagine looking straight at the face you are turning - that face's clockwise defines the positive direction.