Lattice = Points in 3-d space. If all points can be reached by a cerntain vector, then that vector is a primitive lattice vetor. And you have three of those. In your books, this should be covered, and also nice figures. Do the comparison with basis vectors in coordinate system. The lattice points are arranged by translation with the primitive lattice vectors. Then you also have rotation and mirroring, so that you are left with a certain number of lattices that are invariant under rotation and mirroring; and these lattices are called bravais lattices (but some of the baravais lattacies can be reduced even further, but some; for example bcc; is regarded as a bravais lattice since many real crystals have this lattice.
Then you have unit cells and primitive cells. A primitive cell contain ONE lattice point, and you can construct that in an infinite number of ways. Note; a lattice point should be regarded as a sphere, that can be divided by other primitive cells, but the sum of all this is equal to ONE in a primitive cell. See for example a simple cubic primitive cell with 8corners. Each corner is shared amog 8 other cells, so in one primitive cell you have 8*(1/8) = 1 lattive point. Then you have unit cells; for example the bcc cell; that is a cube with one lattice point in the middle ( at ½½½ coordinate).
The thing is to practice, try drawing each unit cell, and play with the vectors, and do excersice in finding basis and bravais lattices etc. Best (only) way to learn.
Then you have the basis, I leave that subject to another volunteer =)