How do I calculate lattice points per unit volume?
Atoms on the corners, edges, and faces of a unit cell are shared by more than one unit cell.  

An atom on a face is shared by two unit cells $\implies$ half an atom belongs to each unit cell.



An atom on the edge is shared by four unit cells



and an atom on the corner is shared by eight unit cells



What does this have to do with the equations?

Well lets say we have an sodium atom that is crystallized in a simple cubic unit cell, then there would be an sodium atom on each of the eight corners of the cell. However, only $\frac{1}{8}$ of these atoms can be assigned to a given cell. Hence the simple cubic structure is 

$$8 \, \text{(corners)} \times \frac{1}{8}=1 \, \text{atom}$$

if sodium formed a body centered cubic structure, there would be two atoms per unit cell (as seen in the top diagram). It isn't shared with an other cells. Therefore the body-centered cubic structure is given by

$$8 \, \text{(corners)}\, \times \frac{1}{8} + 1 \, \text{(body)}= 2 \, \text{atoms}$$

now finally if sodium was a face-centered cubic structure the six atoms on the faces of the unit cell would contribute three net sodium atoms, for a total of four atoms per unit cell. Hence the face-centered cubic structure is

$$8 \, \text{(corners)} \times \frac{1}{8} + \Big{(}6 \, \text{(faces)} \times \frac{1}{2}\Big{)}= 4 \, \text{atoms} $$

Hope this helps