Are there non commutative rings with no zero divisors? If there are,
Are there unity (but not division) rings of this kind?
Are there non-unity rings of this kind?
Sorry, I forgot writting the non division condition.
 A: Yes, in fact much more can be said: There are rings with $1$ such that every non-zero element has a multiplicative inverse. These rings are called division rings or skew-fields.
The real quaternions are an example of a non-commutative division ring.
A: Yes. In fact not only can we force every nonzero element to not be a zero divisor, we can force every nonzero element to be invertible. You get a division ring (also called skew field).
Perhaps one of the most famous historical examples: the quaternions. They are defined by
$$\Bbb H=\{a+bi+bj+ck:i^2=j^2=k^2=ijk=-1,a,b,c,d\in\Bbb R\}.$$
As for nonunital rings: simply take a unital ring with no zero divisors at hand (say $\Bbb H$) and look at a non-unital subring. For example the elements $2\Bbb Z+2i\Bbb Z+2j\Bbb Z+2k\Bbb Z\subseteq\Bbb H$.
A: Take the "polynomials" with integer coefficients in two non-commuting variables $x$ and $y$. If you don't want a unit, use even integers only.
A related example replaces integer coefficients by coefficients in $\mathbb{Z}_2$. 
A: Another kind of example: non-commutative deformations of commutative integral domains, e.g. $\mathbb C[x,y]$ with the commutation relation $[x,y] = 1$.  
