The following is for the one-particle case (i.e. a system with one particle).
Mathematically psi is a function on space, expressed as \psi(x,y,z) and it is complex-valued. Mathematically \psi : \mathbb R^3 \to \mathbb C. Its minimal physical meaning is that its modulus squared, mathematically | \psi |^2, gives the probability density of finding a particle at a certain point in space.
I know that Ĥ is the total of all the energies in a system.
This is true,
in a sense, but I think it's prone to misconception. It is true that
knowing \hat H is equivalent to
knowing all the energy levels of the system (this is known in mathematics as the spectral theorem for operators) but that is not what \hat H
is itself. It
is an operator meaning that is a function which takes a function like psi as its argument/its input, and gives another such function as its image/output. A very simple Hamiltonian (that is its name) is \hat H = -\frac{\partial^2}{\partial x^2} (ignoring constants and expressing it in the one-dimensional case). As input it takes a function \psi and as output it gives minus its second derivative with respect to x, this is of course a new function from \mathbb R^3 to \mathbb C.
E is the specific energy level your system has (in other words you have to choose one; the idea is that you choose the energy level(*) of your system and then calculate what \psi satisfies that equation, and then its modulus squared gives the probability density as discussed above).
(*) Note that you can't choose just any value for E; the possible list is determined by \hat H, as you said.
EDIT: yes, another name for \psi is the so-called "quantum state". Also called "the quantummechanical wavefunction".