When and how do you represent a two body state as a tensor product? I have read that in quantum mechanics, compound systems are constructed as tensor products.
But on page 177 of Griffith, for example, a two body wavefunction is introduced as 
Psi (x1,y1,z1,x2,y2,z2,t).  (Six space dimensions plus time) This is clearly a direct sum, not a tensor product.
If this is correct, when do you use a direct sum and when do you use a tensor product?
Why isn’t the tensor product representing the two body system  
Psi(x1x2,x1y2,x1z2,y1x2,y1y2,y1z2,z1x2,z1y2,z1z2,t) or something similar?
(nine tensor dimensions plus time).  What am I misunderstanding?
And what are the tensor space dimensions?
Or how do these abstract dimensions relate to physical lab space dimensions?
 A: The composition law for quantum systems is always a tensor product. Your problem arises from a confusion over what the tensor product is applied to: you are trying to tensor product the spatial coordinates together, when it is in fact the basis vectors of the Hilbert space you should be tensoring together.
More formally, take two quantum systems A and B, with corresponding Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$. The state of the joint system (A & B) lives in the tensor product space $\mathcal{H}_{AB} = \mathcal{H}_A \otimes \mathcal{H}_B$. This is a fancy-pants way of saying that the basis vectors of the combined Hilbert space look like
$$|\phi\rangle = |A\rangle\otimes|B\rangle.$$
A general state can therefore be written as 
$$|\psi\rangle = \sum\limits_{jk} c_{jk} |A_j\rangle \otimes |B_k\rangle, $$
where the $\{|A_j\rangle\}$ and $\{|B_k\rangle\}$ span Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$ respectively, and $c_{jk}$ are complex numbers. As you would expect, if the two Hilbert spaces have dimension $d$ (if they are spanned by $d$ basis vectors each) then the combined Hilbert space has dimension $d^2$. A general $N$-body Hilbert space has dimension $d^N$.
If the two quantum systems are the position degrees of freedom of two particles, then the total wavefunction looks like
$$\Psi = \sum\limits_{jk} c_{jk} \psi_j^{(1)}(x_1,y_1,z_1,t)\psi^{(2)}_k(x_2,y_2,z_2,t),$$
which clearly depends on 6 spatial coordinates plus time. 
A: A two-body state $|\Psi_{12}\rangle$ can be written as a tensor product $|\Psi_{12}\rangle = |\Psi_1\rangle \otimes |\Psi_2\rangle$ only when there are not correlations between both bodies: e.g., two separated non-interacting bodies. The same happens in classical mechanics, where the two-body state factorizes as $\rho_{12} = \rho_1 · \rho_2$ only in absence of correlations.
The Hilbert space is not the ordinary space and the quantum mechanical variables $(x_1, x_2)$ associated to the position basis $|x\rangle$ do not correspond to any point in ordinary space. Again the same happens in classical mechanics with the classical variables $(x_1, x_2)$ associated to the two-body configuration space.
The physical lab space dimensions are obtained in the one-particle limit.
