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I know what's a basis for some vector space $V$: a set of objects from that space that span the whole space.

We can change between basis by using the change of basis matrix. Basically this matrix transforms a vector representation with respect to a basis to another representation with respect to a new basis.

I'm now wondering what's the relation between a basis of a vector space and a coordinate system for that same vector space?

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    A basis is **not** what you say it is as "the set of ""objects"" in that space" (i.e., the set of **vectors**) must be linearly independent besides being a generator of the whole space. Choosing a basis is the same as choosing a set of coordinates for the space, and every vector's coordinates is the column (or row) n-dimensional vector (with $\;n=\dim V$) of coefficients of the vector when represented as linear combination of the basis elements – 2017-01-14
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    @DonAntonio Yes, I forgot to write the important detail of "linear independence" between the vectors in the basis. Yes, I know the other details too, sorry for not having explicitly said those important details. – 2017-01-14

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In a (finite dimensional) vector space (over a field), one cannot talk about coordinates without referring to a basis.

Given an ordered basis, you have a corresponding "coordinates system" by definition:

Given a basis of a vector space $V$, every element of $V$ can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components.

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    So, as I was suspecting, a basis $B$ for a vector space $V$ is or represents indeed a coordinate system for $V$. – 2017-01-14
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    A basis and the "coordinate system" are two different mathematical objects. One can say that a basis $B$ *induces* a "coordinate system" but one would not say that a basis *is* a coordinate system. "Represents" sounds rather strange too since it implies that one has a "coordinate system" in the first place, which does not make sense. – 2017-01-14
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    Ok, but then what's the difference between a coordinate system for a vector space and the corresponding basis for the space vector space? What does a coordinate system have that the basis does not have? Indeed, this was my original doubt. – 2017-01-14
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    What book are you reading and where did you come up with the question? Everything comes from *definitions*. – 2017-01-14
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    Actually I was reading a lot of things around about computer graphics and I was trying to associate their terminology to mathematics, specifically linear algebra. – 2017-01-14