What is the difference between $|0\rangle $ and $0$? What is the difference between $|0\rangle $ and $0$ in the context of $$a_-  |0\rangle =0~?$$
 A: You may consider 0 as an eigenvalue and write $a|0\rangle = 0|0\rangle$.
Any eigenvector $a|\alpha \rangle = \alpha |\alpha \rangle$ is of different "length" than the corresponding normalized vector $|\alpha \rangle$. In your particular case the vector $0|0\rangle$ is of zeroth length.
A: $|0\rangle$ is just a quantum state that happens to be labeled by the number 0. It's conventional to use that label to denote the ground state (or vacuum state), the one with the lowest energy. But the label you put on a quantum state is actually kind of arbitrary. You could choose a different convention in which you label the ground state with, say, 5, and although it would confuse a lot of people, you could still do physics perfectly well with it. The point is, $|0\rangle$ is just a particular quantum state. The fact that it's labeled with a 0 doesn't have to mean that anything about it is actually zero.
In contrast, $0$ (not written as a ket) is actually zero. You could perhaps think of it as the quantum state of an object that doesn't exist (although I suspect that analogy will come back to bite me... just don't take it too literally). If you calculate any matrix element of some operator $A$ in the "state" $0$, you will get 0 as a result because you're basically multiplying by zero:
$$\langle\psi| A (a_-|0\rangle) = 0$$
for any state $\langle\psi|$. In contrast, you can do this for the ground state without necessarily getting zero:
$$\langle\psi| A |0\rangle = \text{can be anything}$$
A: $|0\rangle$ is a particular nonzero vector in the Hilbert space associated with this system. That vector is nonzero -- in fact, it's usually normalized to have magnitude 1. The 0 on the right refers to the zero vector in the Hilbert space. So they're quite different. For one thing, $|0\rangle$ is a possible state for a particle to be in. 0 isn't (since only unit-magnitude vectors are possible states).
A: $0$ is the additive identity of the vector space, i.e. the element of the vector space which satisfies $$0 + v = v$$ for all $v$ in the vector space. No system can ever be in this state since it is patently unnormalizable.
$|0\rangle$ is the name of an energy eigenstate of some Hamiltonian operator $H$ with the lowest eigenvalue in its spectrum. For example, for the harmonic oscillator $|0\rangle$ corresponds to the gaussian function $~e^{-x^2} $ while "$0$" actually corresponds to the real number zero. For a two-state system "$0$" would correspond to the column vector $\begin{bmatrix}0\\0\\ \end{bmatrix}$.
