I intuitively know what an isomorphism is, but I have a few questions:
I'm looking for answers from a linear algebra point of view. I don't know anything about category theory.
On
To add an example to Matt's answer: the isomorphism between two vector spaces of the same finite dimension over the same field $\mathbb{K}$ is hardly natural, or unique.
Indeed, it's usually produced chosing a basis $v_1, \dots , v_n$ of one of your vector spaces (say, $V$), and since then you can write any vector $v \in V$ as a (unique) linear combination like
$$ v = x_1 v_1 + \dots + x_n v_n \ , $$
you define your isomorphism $\varphi_V: V \longrightarrow \mathbb{K}^n$ as
$$ \varphi (v) = (x_1, \dots , x_n) \ . $$
So far, so good. Now, for any other vector space $W$ of the same dimension over $\mathbb{K}$, you get a direct isomorphism just composing
$$ \phi_{VW} = \varphi_W^{-1}\circ \varphi_V : V \longrightarrow W \ . $$
But this isomorphism $\phi_{VW}$ is not natural, nor canonical, because you need to start by chosing bases for every vector space to define it.
Indeed, if you have a linear map $f: V_1 \longrightarrow V_2$, since you have no control about what $f$ will do with your chosen basis in $V_1$ (if $f$ is not an isomorphism it won't transform your basis in another basis, for instance), there is no chance you can get a commutative diagram like Matt's one. In fact, you don't even have the equivalent of $(f^t)^t$ to check any commutativity.
Neither you have uniqueness: for every basis you chose, the isomorphism $f_V$ changes.
Without getting very categorical, the linear algebra answer to the first two questions is that an isomorphism is natural (or canonical) if it does not depend on the choice of a basis.
There are usually infinitely many isomorphisms between two isomorphic vector spaces, but a natural or canonical one is the unique one satisfying some desirable property.
As an example, let $V$ be a finite-dimensional vector spaces, and let $e_V\colon V\to (V^*)^*$ be the linear map given by $e_V(x)(f)=f(x)$. This is an isomorphism, and the $e$s form the unique collection of isomorphisms such that the diagram
$$\begin{matrix}V&\stackrel{e_V}{\to}&(V^*)^*\\\scriptstyle{f}\downarrow&&\downarrow\scriptstyle{(f^t)^t}\\W&\stackrel{e_W}{\to}&(W^*)^*\end{matrix}$$
commutes for any finite-dimensional vector space $W$ and linear map $f\colon V\to W$. (Here $f^t$ denotes the transpose of $f$). (As a categorical aside for those who might appreciate it; the word natural is arising here because the collection of isomorphisms $e_V$ provide a natural transformation between the identity functor and the double dual functor on finite dimensional vector spaces over some fixed field).
The working dictionary should probably be that natural isomorphisms don't depend on choosing bases, and canonical ones are unique isomorphisms with some additional property, but these two things usually coincide.
To prove two vector spaces are isomorphic, one must write down an isomorphism - or apply some theorem that implies the existence of one. For example if $V$ and $W$ are vector spaces of the same finite dimension over the same field, then they are isomorphic.