Let C be an ascending chain of linearly independent sets, each set containing an independent vector that was not present in the set before. If a linear combination of vectors drawn from any of these sets is 0, those vectors are all present in one of the sets in C. However, that set is linearly independent. Therefore the union of the entire chain of sets is linearly independent. Use zorn's lemma to assert the existance of a maximal linearly independent set. This maximal set of independent vectors is a basis. Remember that a basis might be infinite, perhaps uncountably infinite.
You can start the chain with 0, or you can start with a set of independent vectors b0 and consider chains of sets that contain b0. Now a maximal set is a basis containing b0.
Suppose s is a set of vectors that forms a basis, and y is a vector that is not spanned by s. In other words, y is not a linear combination of vectors taken from s. Since s is maximal, y is a dependent vector. Some combination of y and vectors from s produces 0. If j is the coefficient on y, multiply through by j inverse on the left, and y becomes a linear combination of vectors from s after all. (This works over a division ring, as well as a field.) Therefore each basis spans the entire space. Nothing is left out. Conversely, an independent set of vectors that spans the entire space is maximal, and is a basis.
Given a basis, each point has a unique representation relative to that basis, i.e. each point is a unique linear combination of basis elements. If there are two representations, subtract them to produce a linear combination of basis elements that equals 0, which contradicts linear independence.
A vector space over the ring R is isomorphic to the direct product of copies of R. A coefficient from R is applied to the first vector, and independent of that, a coefficient is applied to the second vector, and a coefficient is applied to the third vector, and so on. When a vector is scaled by k, all its coefficients are scaled by k. When two vectors are added together, their coefficients are added together, coordinate by coordinate. The vector space acts just like so many copies of the ring R, running in parallel. This is called a free R module.
If the coefficients are real numbers, the vector space looks like a line, the plane, 3 space, or some higher dimensional space. Consider a 3 dimensional example. Select a basis and call the basis vectors r s and t. Points in the vector space are now assigned "coordinates" relative to this basis, and the vector space looks just like "real space". We can talk about the point 7r+3s-5t, and map this back to coordinates in x y and z, which is yet another basis. The x y z basis is standard, but for most applications any basis will do. (Mutually perpendicular basis vectors, like x y z, do offer some advantages.)