Primitive cells and their variations
I will not consider hexagonal cell in this answer.

Unit cell is generally parallelepiped, which has three sides $a$, $b$ and $c$ and three angles between them $\alpha$, $\beta$ and $\gamma$. The sides can be f.e. of equal length and angles can be same and equal to $90^\circ$, this way you will have cubic unit cell. You can try to vary sides and angles and you will discover, there is only those seven types of unit cells.

When I way learning this I found useful third chapter in "Foundations of material science and engineering". At least there is nice table and images to show those cells. You can find it by typing "Crystal structures and crystal geometry" into Google.

Well, now answer last question. Let's say we have cubic base-centered (more correct than end-centered, I think) cell. From top it would look like this:



I draw only two cubic base-centered cells (I can add more if it will be more clear) next to each other. However, as you can see, there is one smaller cell (red) inside of our base-centered lattice (green) and that is simple tetragonal cell (If you draw it in three dimension, you will see, that the height is $\sqrt{2}$ times greater). As it has lower volume, this is the primitive cell, not that base-centered.