Where does the notion of pseudoholomorphic curve come from? I wonder, why we consider the notion of pseudoholomorphic curve: By definition a pseudoholomorphic curve in an almost complex manifold $X$ is a smooth map $f: C \rightarrow X$
from a Riemann surface $C$ into $X$ such that $df \circ J=J \circ df$ for the respective almost complex structure $J$. Why does it make sense to look at this definition? Is it just because the Cauchy-Riemann differential equations $\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$, $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
are invariant under the symmetry $x \rightarrow y$, $y \rightarrow -x$ and $u \rightarrow v$, $v \rightarrow -u$ and therefore we have that this relation holds for a holomorphic map $f$ or is there any other reason?
 A: I think it would not be wrong to say that pseudo-holomorphic curves became really popular thanks to the work of Gromov Pseudo-holomorphic curves in complex manifolds http://www.ihes.fr/~gromov/PDF/9[45].pdf
In this work Gromov shows that certain facts about holomorphic curves in complex manifolds survive when the holomorphic structure is replaced by merely an almost complex structure provided the almost complex structure is tamed by a symplectic one.
For example there is a statement that every mathematician knows - for every two points in a plane there is a unique line that passes through them. This statement survives and becomes a difficult theorem in symplectic geometry that helps in particular to classify symplectic structures on $\mathbb CP^2$.
Pseudo-holomorphic curves were appearing here and there also without (an apparent) relation to Gromov's work. For example Elles-Salamon noticed that minimal surfaces in hyperbolic 4-space are in correspondence with almost complex curves in the twistor space of the hyperbolic space:
J. Eells and S. Salamon. Twistorial construction of harmonic maps of
surfaces into four-manifolds.
A: The work of Floer in proving the Arnold conjecture (at first in the monotone setting) interpreted pseudoholomorphic curves as a generalization of the flowlines in Morse theory.  In analogy with Morse homology, a count of pseudoholomorphic curves provides the differential in what has become known as a Floer homology.
This work predated Gromov's paper and has been seminal in its own way, spawning a variety of Floer homologies (such as the popular Heegaard-Floer homology) which provide strong invariants in low-dimensional topology.
If you're willing to accept that symplectic manifolds are reasonable things to study, then Floer's approach to the Arnold conjecture shows pseudoholomorphic curves arising very naturally.  Salamon has a nice paper explaining this carefully http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf .
A: As many have pointed out: Gromov introduced it* wrote a seminal paper utilizing it, and we continue to use it today because it's an incredibly useful tool. I've never spoken to Gromov about why he introduced it (who knows how great mathematicians come up with great ideas) but I can try to give some (probably historically false) motivations as to why someone might have come up with the notion. For instance, if Gromov hadn't discovered it, you might have come up with it as follows:
(1) First, complex geometry--if you like, you can think of algebraic geometry--has a lot of rigidity. The fact that we can even give a discrete count to sub-objects (like how many curves pass through n fixed points) is special -- the question takes on a totally different nature in more flimsy geometries.
Now, is there a way to relax the background of complex geometry, and still come up with a useful, fun theory? For instance, how necessary is the integrability condition on J (the complex structure) to still make sense of curve-counting?
What Gromov showed is that if the complex structure is `tame' in the sense that one has a compatible symplectic form, questions about curve-counting can still have nice answers. Really, the difference between a pseudoholomorphic curve and a holomorphic curve isn't in their definitions, it's in the nature of J in the target. Relaxing the J from "integrable complex structure" to "complex structure tamed by a symplectic form" is the generalization that's happening.
(1') Put another way, we already had a famous 2-out-of-3 principle recognizing the relationship between Riemannian, complex, and symplectic structures on a vector space. Studying curves on complex projective varieties take on rigidity, in some sense, because we study maps between manifolds with Kahler structure: manifolds both symplectic and complex, and further, each structure is integrable--in that the Nijenhaus tensor vanishes, and omega is closed. It's natural to ask whether we can still find interesting structure in the 2-out-of-3 world by studying manifolds whose tangential structures are compatibly Riemannian, complex, and symplectic, but which do not satisfy a global condition like integrability of J or closedness of $\omega$. And when you get rid of the integrability of J, it turns out that you can find such a structure on any symplectic manifold. (In fact, once you fix $\omega$, there's a contractible space of compatible $J$. That's why pseudoholomorphic curves can be applied widely in the symplectic world.)
(2) There might be another motivation from physics. In mirror symmetry, one predicts the existence of mirror Calabi-Yau manifolds. A field theory that relies on the symplectic structure of one manifold should correspond to a field theory that relies on the complex structure of the mirror. And the correlation functions count J-holomorphic curves in the symplectic manifold. Historically though, I'm not sure if physics alone would be able to motivate the study of these field theories on just symplectic manifolds with almost-complex structure, as opposed to Calabi-Yaus. Somebody with more background could probably comment on this.
*As I learned from Antoine and Dmitri, there were previous works utilizing pseudo-holomorphic curves. For instance: 
A.Nijenhuis, W.Wolf. Some integration problems in almost-complex and complex manifolds, Ann. Math. 77 (1963), 
J. Eells and S. Salamon. Twistorial construction of harmonic maps of surfaces into four-manifolds. (1985).
