To elaborate a bit on Tobias answer. The notion of isomorphism depends on which structure (category actually) you are studying.
Edit: Pete L. Clark pointed out that I was too sloppy with my original answer.
The idea of an isomorphism is that isomorphisms preserve all structure that one is studying. This means that if $X,Y$ objects in some category, then there exists morphisms $f:X\rightarrow Y$, $g:Y\rightarrow X$ such that $f\circ g$ is the identity on $Y$, and $g\circ f$ is the identity on $X$.
To be a bit more explicit, if $X$ and $Y$ are sets, and there is a bijective function $X\rightarrow Y$, then we can construct the inverse $f^{-1}:Y\rightarrow X$. This inverse function is defined by $f^{-1}(y)=x$ iff $f(x)=y$. We have that $f\circ f^{-1}=id_Y$ and $f^{-1}\circ f=id_X$.
But if we are talking of vector spaces, we demand more. We want two vector spaces to be isomorphic iff we can realize the above situation by linear maps. This is not always possible, even though there exists a bijection (you cannot construct a invertible linear map $\mathbb{R}\rightarrow \mathbb{R}^2$). In the linear case; if a function is invertible and linear, its inverse is also linear.
In general however, it need not be the case that the inverse function of some structure preserving map preserves the structure. Pete pointed out that the function $x\mapsto x^3$ is an invertible function. It is also differentiable. but its inverse is not differentiable in zero. Thus $x\mapsto x^3$ is not an isomorphism in the category of differentiable manifolds and differentiable maps.
I would like to conclude with the following. We cannot blatantly say that two things are isomorphic. It depends on the context. The isomorphism is always in a category. In the category of sets, isomorphisms are bijections, in the category of vector spaces isomorphisms are invertible linear maps, in the category of groups isomorphisms are group isomorphisms. This can be confusing. For example $\mathbb{R}$ can be seen as lot of things. It is a set. It is a one dimensional vector space over $\mathbb{R}$. It is a group under addition. it is a ring. It is a differentiable manifold. It is a Riemannian manifold. In all these $\mathbb{R}$ can be isomorphic (bijective, linearly isomorphic, group isomprhic, ring isomorphic, diffeomorphic, isometric) to different things. This all depends on the context.
The terminology "kernel" and "nullspace" refer to the same concept, in the context of vector spaces and linear transformations. It is more common in the literature to use the word nullspace when referring to a matrix and the word kernel when referring to an abstract linear transformation. However, using either word is valid. Note that a matrix is a linear transformation from one coordinate vector space to another. Additionally, the terminology "kernel" is used extensively to denote the analogous concept as for linear transformations for morphisms of various other algebraic structures, e.g. groups, rings, modules and in fact we have a definition of kernel in the very abstract context of abelian categories.
Best Answer
For any algebraic structure, a homomorphism preserves the structure, and some types of homomorphisms are:
Note that these are common definitions in abstract algebra; in category theory, morphisms have generalized definitions which can in some cases be distinct from these (but are identical in the category of vector spaces).
So a linear transformation $A\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ is a homomorphism since it preserves the vector space structure (vector addition, scalar addition and multiplication, scalar multiplication of vectors), e.g. $A(av+w)=aA(v)+Aw$. It is an epimorphism if its image is $\mathbb{R}^{m}$, a monomorphism if it has zero kernel, an endomorphism if $n=m$, and an automorphism (as well as an isomorphism) if all of these are true.
The below figure might be helpful. More details here.