We use a cartesian coordinate system with origin at the centre of gravity of the whole cube, and three axes x, y, z in right-hand order.
A cube has 26 pieces: 8 corners, 12 edges, and 6 centres.
A cube has 6 faces: Right (+x), Left (-x), Up (+y), Down (-y), Front (+z), and Back (-z). Another way to identify faces is by the colour of their centres (regardless of the colours on other pieces).
"Face" is a 2-dimensional concept and "side" is its 3-dimensional counterpart, so a side consists of 9 pieces in all their volume.
Periphery is the part of a side that is not part of the corresponding face.
A sticker is one of the colored flat surfaces of a piece that are visible from the outside.
A layer is a collection of 9 pieces that are on the same level with respect to a given axis. For each of x, y, and z, there are three layers: two sides and the middle layer (a.k.a. middle slice) sandwiched between them.
A twist is the rotation of a corner or an edge. A corner can be twisted in 3 different ways, and an edge in 2.
A turn is the rotation of a side.
A move is a sequence of rotations of sides, middle layers, or of the whole cube.
An algorithm is a move that achieves a particular purpose, like twisting or permuting a set of pieces.
L R U D F B # left,right,up,down,front,back side clockwise l r u d f b # wide moves - side and middle layer together x y z # whole-cube rotations M E S # middle layers α2=αα; α3=ααα; # repeat 2,3.. times α' # inverse α! # left-right mirroring [α:β]=αβα' # conjugate [α,β]=αβα'β' # commutator (α) # parentheses for grouping $=[R,U]=RUR'U' # sexy move %=[R',F]=R'FRF' # sledgehammer @=RUR'URU2R' # sune