Unit Cell: this is the smallest repeating unit in crystal. Each of the structures described on this site will have a the associated unit cell to show the arrangement of the atoms, but for now we should familiarise ourselves with a 2D unit cell. It's worth noting at this point that opposite sides of a unit cell must be the same so when a translation occurs the atom at the 0 position will sit on an identical one at the 1 position. If a position in the unit cell can be found where the atoms are the same in all direction then the unit cell is valid. This point is know as the lattice point.
Imagine some red and yellow balls in a large triangle, the unit cell for this arrangement of balls would six balls that form a rhombus(on the diagram one of the groups is outlined in blue). the unit cell can translate one whole length of the cell in any direction and will overlay an identical array original image. On first inspection it may appear that the unit cell could be just four balls-2red and 2yellow...however this is not the case as the atoms at each of the corners would not be the same and so a lattice point could not be defined.
It is often possible to define more than one unit cell for a structure, or in this case arrangement of balls. The ball at position 0 could have been a yellow ball, so long as the balls at 1 were also yellow. Similarly we could have double the length of each side, and a unit cell would be made, but convention selects the smallest possible unit cell.

Atomic Co-ordinates: if we were to superimpose a unit cell onto a graph we could describe the location of the atoms or ions using the coordinates of the graph, either 2-D or 3-D. The origin of the graph being (0,0,0), and the atom at the furthest distance from the origin being at (1,1,1) and all other points in between being described using fractions of the distances. So an atom at the centre of a cube would have the atomic coordinates (1/2,1/2,1/2), since it is halfway along each axis of the unit cell. What would the coordinates of the atoms on the following diagram be? [add picture and answers]

Direction: When talking about directions in crystals and lattices we refer to the displacement in each plane X, Y and Z. The physical distance between the atoms does not matter when talking about direction. The directions are written as [X,Y,Z]. For those familiar with vectors this may need no further explanation. For clarity though, the [1,1,1] direction moves an equal distance in both the the x, y and z planes. This movement would correspond to going to opposite vertices of a cube.[ADD PIC!?!?!?!?!]
the [0,0,1] direction is straight along the Z axis, since there is no displacement described along either x or y. The values of x y and z do not have to be whole numbers. hence [1/2,1,1/2] is a valid direction. Its worth realising here that [1/2,1/2,1/2] is more easily written as [1,1,1], what is important is that the relative size of x y and z remains the same. Values of X, Y and Z are always positive, since if a negative value was included the direction could be described in a different manner where all positive values are quoted.

Atomic Planes: in a crystal planes are defined using 3miller indices, in a similar method used for cell directions. A point on each of the x y or z axis are all passed through by a plane, the position of these are described using miller indices. These indices are h k and l, written (h,k,l), which are the reciprocals of the points of interception on x, y and z. For example if the X axis is intercepted at 0.5 by the plane the Miller indice, h, would be 2. All planes with the same value of h k and l are parallel, similarly if the values of h k and l are in the same ratio the planes are still parallel. All of these planes would therefore be parallel: (121), (121), (242),(484)

Projections:a projection is simply an end on view of the unit cell. Imagine a cube with a ball at each of the 8 vertices. If you looked at the cube from directly above you would only see the four spheres nearest you, but you know the other four are directly behind. A projection of this could be drawn like below.