Calabi - Yau Manifolds

There are several different "definitions" of Calabi-Yau manifolds, not all equivalent, and not all contained in one general definition. A good discussion of some of these inequivalent definitions can be found in Joyce's book:

Compact Manifolds with Special Holonomy

The other answers you have gotten so far seem to be from the algebraic geometry side of things, and are fine in that context. From a Riemannian geometry point of view, the most natural definition of a Calabi-Yau manifold (whether compact or non-compact) is a $2n$-dimensional Riemannian manifold for which the holonomy of the Levi-Civita connection is exactly $SU(n)$. Allowing the holonomy to be a proper subgroup of $SU(n)$ is also common. In that case, hyperKahler (which is holonomy $Sp(n/2)$ in dimension $4n$) can also be considered as being Calabi-Yau, for example.

This Riemannian geometry definition is equivalent to the existence of the "Calabi-Yau package": a Riemannian metric $g$, an integrable complex structure $J$ (orthogonal with respect to $g$) together which induce the associated Kahler form $\omega$ by $\omega(X,Y) = g(JX, Y)$, and a holomorphic volume form $\Omega$, which is a holomorphic $(n,0)$-form on $M$. These tensors must satisfy:

(1) $\nabla \omega = 0$ (equivalent to $\nabla J = 0$, the Kahler condition) This is also equivalent to $d\omega = 0$ because we are assuming $J$ to be integrable.

(2) $\nabla \Omega = 0$

(3) $\frac{\omega^n}{n!} = c_n \, \Omega \wedge \bar \Omega$ for some universal constant $c_n$ depending only on the dimension which I can never remember.

These conditions imply, in particular that $g$ is Ricci-flat and $c_1(M) = 0$. Also, if the holonomy is exactly $SU(n)$ rather than a proper subgroup, then it also follows that $h^{p,0} = h^{0,p} = 0$ for all $1 \leq p \leq n-1$.

In http://en.wikipedia.org/wiki/Calabi_Yau_manifold there is a discussion of some of the many definitions of CY manifold and the relations between them. Yau defines them in http://www.scholarpedia.org/article/Calabi-Yau_manifold as "compact, complex Kähler manifolds that have trivial first Chern classes (over R). In most cases, we assume that they have finite fundamental groups." The strictest definition would also require vanishing integral Chern class.

For singular varieties , Siu introduced Calabi-Yau varieties by using numerical Kodaira dimension

Let $X_0$ be a projective variety with canonical line bundle $K\to X_0$ of Kodaira dimension $$\kappa(X_0)=\limsup\frac{\log \dim H^0(X_0,K^{\otimes \ell})}{\log\ell}$$ This can be shown to coincide with the maximal complex dimension of the image of $X_0$ under pluri-canonical maps to complex projective space, so that $\kappa(X_0)\in\{-\infty,0,1,...,m\}$. Also since in general we work on Singular K\"ahler variety we need to notion of numerical Kodaira dimension instead of Kodaira dimension.

$$\kappa_{num}(X)=\sup_{k\geq 1}\left[\limsup_{m\to \infty}\frac{\log\dim_{\mathbb C}H^0(X,mK_X+kL)}{\log m}\right]$$

where $L$ is an ample line bundle on $X$.Note that the definition of $\kappa_{num}(X)$ is independent of the choice of the ample line bundle $L$ on $X$. Siu formulated that the abundance conjecture is equivalent as $$\kappa_{kod}(X) =\kappa_{num}(X)$$

The Minimal Model and abundance conjectures would imply that every variety of Kodaira dimension $\kappa (X)=0$ is birational to a Calabi-Yau variety with terminal singularities.

Note that when $K_X$ is psudoeffective vanishing numerical Kodaira dimension is equivalent that $X$ is Calabi-Yau variety