I have the following question: If $E$ is a ring spectrum, then a complex orientation of $E$ is an element of $E^2(\mathbb{C}P^{\infty})$ that is mapped to $1$ in $E^2(\mathbb{C}P^{1})$.
I have read that there is a natural orientation on $MU$ the spectrum of complex bordism. Can you tell me, how this orientation is defined and why it is an orientation?



Orientation of complex bordism spectrum
A complex orientation on a homotopy commutative ring spectrum $E$ is sometimes defined to be an element in $\widetilde{E}^2(\mathbb{CP}^\infty)$ which restricts to the canonical element of $\widetilde{E}^2(S^2) = \pi\_0 E $. 


In fact, there is another definition of an orientation, which generalizes better to other settings: a complex orientation on $E$ is an assignment of a Thom class for each complex vector bundle $V \to X$ (that is, a class in $\widetilde{E}( B(V), S(V))$ which restricts to the canonical generator on each copy of $(B(V\_x), S(V\_x)) \subset (B(V), S(V))$ for each $x \in X$. (In ordinary homology, this uniquely determines the Thom class, but in generalized homology, this is *additional data* rather than a merely condition to be satisfied.)


For a complex orientation, this assignment is required to be functorial, and moreover multiplicative: Thom classes for vector bundles $V, W \to X$ determine a Thom class for $V \oplus W$ (because the Thom space of $V \oplus W$ is the smash product of the Thom spaces of $V, W$). If you think about it this way, it is easier to see that a complex orientation on $E$ is the same as a map of ring spectra $MU \to E$. In fact, a complex orientation on $E$ means that the *universal* vector bundle $\xi\_n \to BU(n)$ has a complex orientation, which if you unwind is a map $\mathrm{Th}(\xi\_n) \to E$. These give the spaces in the $MU$-spectrum and multiplicativity makes this a map $MU \to E$ (strictly speaking, one has to make some kind of a $\lim^1$-argument to make this rigorous). 


One reason that this is a more natural definition is that it generalizes to other types of bundles than complex bundles: one can talk about a real-oriented cohomology theory $E$ (which comes from a map $MO \to E$) or a spin-oriented cohomology theory (coming from a map $M \mathrm{Spin} \to E$), among other things. It takes a little work, though, to show that complex orientability is equivalent to the degeneration of the AHSS for $\mathbb{CP}^\infty$, but one direction is easy: if $E$ has a complex orientation in the sense of this answer, then we get a class in $\widetilde{E}^2(\mathbb{CP}^\infty)$ by regardeding $\mathbb{CP}^\infty$ as the Thom space of the tautological bundle over itself. 


You can read more about this in the [course notes](http://people.virginia.edu/~mah7cd/Foundations/coctalos.pdf) for "Complex oriented cohomology theories and the language of stacks," among other places. I also found Jacob Lurie's notes on "[chromatic homotopy theory](http://www.math.harvard.edu/~lurie/252x.html)" very helpful. 


