Any non-singular complex variety $V$ of dimension $n$ (in either affine space or projective space) can be endowed with the structure of a complex manifold of dimension $n$. Moreover as a submanifold of a Kahler manifold, it will also be Kahler.
The passage to a real manifold can be slightly subtle, as there are two approaches you can take. Firstly, you can simply consider the complex manifold as a real manifold of dimension $2n$, with a complex structure.
Alternatively, if the variety is defined over $\mathbb{R}$ (that is, it has equations with real coefficients), then you can look at the set of real points $V(\mathbb{R})$, which you can also endow with the struture of a real manifold. As a submanifold of a Riemannian manifold it is also a Riemannian manifold.
I should note that some funny things can happy happen though, for example even if $V$ is connected then $V(\mathbb{R})$ may not be.
If your variety has singularities, these always occur on a closed subset so you can just remove them to get a non-singular variety. Alternatively you can resolve them as Ariyan suggests.
