The simplest example of a complex analytic non-algebraic manifold (and hence, non-projective) is probably the Hopf surface. Indeed, any smooth complete complex algebraic variety, projective or not, is bimeromorphically Kaehler,so its cohomology admits a Hodge decomposition, which can't exist for the Hopf surface, since its first Betti number is odd.

Any smooth complex algebraic variety is Moishezon, i.e. the transcendence degree of the field of meromorphic functions equals the dimension. All Moishezon surfaces are algebraic and even projective (Kodaira), but starting from dimension 3 there are Moishezon non-algebraic varieties. Here is an example (given in Hironaka's thesis). Let $C$ be a nodal plane cubic (or any other curve with one node and no other singularities) in $\mathbf{P}^3(\mathbf{C})$ and let $P$ be the singular point of $C$. Take a Euclidean neighborhood $U$ of $P$ such that $U\cap C$ is analytically two branches $C_1$ and $C_2$ intersecting transversally. Let $X$ be $\mathbf{P}^3(\mathbf{C})\setminus\{P\}$ blown up along $C\setminus{\{}P{\}}$ and let $Y$ be the result of blowing up $U$ along $C_1$ and then blowing up the result along the proper preimage of $C_2$. Note that $Y$ exists only in the analytic category.

Both $X$ any $Y$ map to $\mathbf{P}^3(\mathbf{C})$ and the parts of both $X$ and $Y$ over $U\setminus P$ can be naturally identified. So we glue them together to get an analytic manifold $Z$. It is Moishezon, since it is bimeromorphic to $\mathbf{P}^3(\mathbf{C})$. Let us show that it is not algebraic. Let $L$ be the preimage of a point in $C\setminus P$ and let $L_i,i=1,2$ be the preimage of a point of $C_i\setminus P$. The preimage of $P$ itself is two transversal lines, $L'$ and $L''$, the first of which appears after the first blow-up and the second one after the second. We have $[L]=[L_1]=[L_2], [L_2]=[L''],[L_1]=[L']+[L'']$. (Here I really wish I could draw you a picture!) So $[L']=0$ i.e. we have a $\mathbf{P}^1$ inside $Z$ which is homologous to zero. This is impossible for an algebraic variety (as David writes in his blog posting).

Hironaka's thesis also contains examples of complete, algebraic but not projective manifolds constructed in a similar fashion.

upd: woops, wrote this answer in a hurry just before going out for drinks; missed a couple of things as a result. These have now been fixed. The curve $C$ is a nodal plane cubic, not conic. Also, David doesn't actually show in his posting that the class of an irreducible curve in a smooth complete variety is non-trivial, but this is easy anyway: let $Z$ be the ambient smooth complete variety and let $W$ be an irreducible curve. Take a smooth point $Q$ of $W$ and let $U$ be an affine neighborhood containing $Q$. There is an irreducible hypersurface $S$ through $Q$ in $U$ that does not contain $W\cap U$ and intersects $W$ transversally at $Q$. So the closure $\bar S$ of $S$ in $Z$ does not contain $W$ and intersects $W$ transversally at at least one point. So the Poincar\'e dual class of $\bar S$ takes a positive value on the class of $W$.

Note that the class of a reducible curve may well be zero.