Difference between "|Z|" and "Resistance" in capacitor datasheet?
One graph says it is displaying |Z|, whereas the other graph says it's displaying "R". Obviously the 2 graphs are different, 


Impedance (\$Z\$) has both resistive and reactive components:

$$Z = R+jX$$

So if the ESR is small compared to the reactance of the capacitor (as it should be, especially for low frequencies) then \$\left|Z\right|\$ has almost nothing to do with ESR, and almost everything to do with \$jX_c = \frac{1}{j\omega C}\$.

At high frequencies, although it isn't a desirable characteristic, the inductive reactance of the part tends to dominate \$Z\$, and again the ESR becomes mostly irrelevant to determining \$\left|Z\right|\$.


  but what's the meaning of the second one? Is it ESR?


Yes, the second curve is giving ESR. This is important in many applications, because even though it doesn't much affect the behavior of the circuit, it does affect how much the capacitor self heats, as well as the Q factor of any resonator using the capacitor as a component.