All quotes will be from Solid State Physics by Ashcroft and Mermin.
A fundamental concept in the description of any crystalline solid is that of the Bravais lattice, which specifies the periodic array in which the repeated units of the crystal are arranged. The units themselves may be single atoms, groups of atoms, molecules, ions, etc., but the Bravais lattice summarizes only the geometry of the underlying periodic structure, regardless of what the actual units may be."
Primitive Unit Cell:
A volume of space that, when translated through all the vectors in a Bravais lattice, just fills all of space without either overlapping itself or leaving voids is called a primitive cell or primitive unit cell of the lattice.
Unit Cell; Conventional Unit Cell:
One can fill space up with nonprimitive unit cells (known simply as unit cells or conventional unit cells). A unit cell is a region that just fills space without any overlapping when translated through some subset of the vectors of a Bravais lattice. The conventional unit cell is generally chosen to be bigger than the primitive cell and to have the required symmetry.
A physical crystal can be described by giving its underlying Bravais lattice, together with a description of the arrangement of atoms, molecules, ions, etc. within a particular primitive cell.
So, one comes up with 14 Bravais lattices from symmetry considerations, divided into 7 crystal systems (cubic, tetragonal, orthorhombic,monoclinic, triclinic, trigonal, and hexagonal). This comes solely by enumerating the ways in which a periodic array of points can exist in 3 dimensions.
Now, what is on those points is a unit cell, which will itself have some symmetry. Thus, the combination of Bravais lattice and unit cell symmetry can again be enumerated and one comes up with 230 space groups.
Now for some of your related questions:
All cubic-related Bravais lattices will have 90 degree angles because they are based on cubic symmetry. The trigonal Bravais lattice has no 90 degree angles, but isn't talked about much in more basic textbooks because, well, it looks weird.
Why no pentagonal unit cells? Well, because you can't fill space with a 5-fold symmetric Bravais lattice. Quasicrystals, while they have 5-fold symmetry, are a tiling through space that does not obey the rules for a Bravais lattice.