From the point of view of complex geometry there is a rough partial hierarchy of compact complex manifold; these are the projective, Moishezon, Kähler and Fujiki manifolds.
A compact complex manifold $X$ is projective if it can be embedded into a projective space $\mathbb P^N$. By Serre's GAGA theorem this means that it is algebraic, i.e., it can be defined as the zero locus of a finite number of polynomials. In diffeo-geometric terms, a manifold is projective if and only if it admits a holomorphic line bundle $L \to X$ with a positively curved hermitian metric. This is Kodaira's embedding theorem.
A larger class of manifolds are the Kähler manifolds. These are $X$ that admit a hermitian metric $h$ (on the tangent bundle), whose Kähler form is closed. This means that the smooth $(1,1)$-form $\omega = - Im h$ is $d$-closed, or $d \omega = 0$. Any projective manifold is Kähler, since the curvature form of a positively curved metric on a line bundle defines a Kähler metric. The converse is false; a Kähler manifold is projective if and only if it admits a Kähler metric $\omega$ whose cohomology class is integral (in which case one can find a line bundle on $X$ such that the metric is the curvature form of a positive hermitian metric by a lemma of André Weil).
A Moishezon manifold can now be characterized as a modification of a projective manifold, so there exists a generically 1-1 meromorphic map $X \to Y$, where $Y$ is projective. This is equivalent to $X$ admitting an integral Kähler current, which is roughly a closed $(1,1)$-form $T$ that has distribution coefficients (in local coordinates) whose cohomology class is integral. We also demand a positivity property: if $\omega$ is hermitian, then for every small $\epsilon > 0$ the current $T - \epsilon \omega$ should be positive.
A Fujiki manifold is then a modification of a Kähler manifold. Similarly, this is equivalent to $X$ admitting a Kähler current $T$, without conditions on its cohomology class. Any Moishezon manifold is thus a Fujiki manifold.
These classes do absolutely not cover all compact complex manifolds. To see this, we remark that the Hodge decomposition holds for all of these classes (this is slightly nontrivial). The Hodge decomposition fails for the Hopf surface, so that general nonprojective (non-Kähler) manifold does not fit into any of these classes. Little is known about general non-Kähler manifolds, but a conjecture of Bogomolov states that they can all be obtained via algebraic constructions on foliations of projective manifolds.
For details on these differential geometric characterizations you could look at Jean-Pierre Demailly's book on complex geometry and his lecture notes on transcendental methods in algebraic geometry (both available for free on his website).