to expand on CaptainBlack's example, one may take physical measurements in 2 different locations. a different "origin" (reference point) may be used at each location (instead of a common reference point like the greenwich observatory).
or one may simply use different units of measurement, resulting in the need for "scaling factors" in the unit vectors.
in the world of acoustics, one may have different oscillators (or circuits) one uses for generating (or processing) sound. so different settings may be required to produce a (nearly identical) sound on differing pieces of equipment.
for studying some kinds of motion (like projectiles) a rectangular coordinate system may be the most convenient, for studying other kinds (like orbits), a polar system may work better. we need to know how to go "back and forth" between the two.
in economics we may note that different economic quantities may vary linearly (or can be approximated linearly), and be linearly independent. depending on what you want to DO with this information, you may want to consider each quantity as a "basis element" or you may want to consider "aggregates" (perhaps prices as a linear combination of various cost factors) instead as a basis (for determining a suitable price index in a certain industry, perhaps).
in field (galois) theory, it may be "natural" to consider two different bases for the same space. for example, it turns out that the field generated by a (primitive) 3rd root of unity over the rationals is the same field generated by a (primitive) 6th root of unity. if we call the 3rd root w, and the 6th root u, we have:
w = u2 or 1/u2
u = -1/w or -w
which means that either the basis {1,u} or {1,w} can be used when dealing with this field.
for more complicated vector spaces, such as the vector spaces of all continuous functions defined on a real interval [a,b], there may be no natural "standard" basis. so depending on which functions in this vector space you are studying (exponentials, polynomials, etc.) certain bases may be more convenient to work with. a fact proved in one basis may indeed "transfer" to another basis, but to do so, you might need "the change of basis" transformation (usually, for large vector spaces such as this one, describing such a transformation by a matrix is not feasible).
the point i want to get across is that linear algebra applies to "much more" than just "euclidean 3-space" (the normal xyz-system you are accustomed to). it's a very useful tool for many areas (ballistics, economics, statistics, manufacturing optimization, signal processing, cryptography, it's a long list...) that we wouldn't normally conceive of as "spaces" (in the geometric sense).
