Definition for pseudoholomorphic curves A pseudoholomorphic curve is a map $u:(\Sigma,j)\to(M,J)$ from a Riemann surface $\Sigma$ with an almost complex structure $j$ to a manifold $M$ with an almost complex structure $J$. We require moreover that u satisfies the "Cauchy-Riemann equations"
$$J\circ du=du\circ j.$$
I would like to know the difference between holomorphic and pseudoholomorphic curves. and why lack of holomorphic objects ?
 A: A good starting point is to first investigate that holomorphic and pseudoholomorphic functions are the same once you chose $(\mathbb{C},i)$ as your Riemann surface and $(\mathbb{C},ds \wedge dt)$ (here $z=s+it)$ as your symplectic manifold. The almost complex structure $J$ is then given again by the multiplication by $i$. Plugging in the definintions, you can see that
$$
J \circ du=du \circ i \iff \partial_{\bar{z}} u=0
$$
i.e. the differential has vanishing antiholomorphic part (and therefore statsifies the Cauchy-Riemann equations).
In general, you cannot assume for $(M,\omega)$ to carry a complex structure, but just an almost complex structure. Even for fairly simple manifolds, it is non-obvious wether there exists an integrable almost complex structure or not. Just take the manifold $S^6$ - this is already problematic and as far as I know an open problem! Therefore, it is not possible to talk about complex/holomorphic structures for maps $u: \Sigma \to M$ in some cases.(Note that it was pointed out correctly in the comments that $S^6$ is not symplectic; however, it still illustrates the point that a complex structure is hard to come by).
The next point is that pseudoholomorphic curves should rather be seen as a generalization of maps between Riemann surfaces in my opinion. A lot of features, such as bubbling, removable singularity theorems, harmonic-holomorphic correspondence carry over from the theory of mapppings between two Riemann surfaces $u:\Sigma_1 \to \Sigma_2$. A crucial role is i.e. taken by the dirichlet energy
$$
E(u):=\frac{1}{2}\int_{\Sigma}|du|^2 dvol_{\Sigma},
$$
which is rather irrelevant in standard complex analysis classes. Note that in this case any almost complex structure is already integrable for dimensional reasons (also see Nijenhuis tensor in this context).
Many properties of mappings between Riemann surfaces, especially local ones, are inherited or related from the standard case of holomorphic functions from $\mathbb{C}$ to $\mathbb{C}$. So in the end you could say that pseudoholomoprhic functions generalize maps between Riemann surfaces which generalize holomorphic functions from $\mathbb{C}$ to $\mathbb{C}$ .
