Why inductive and capacitive reactances have opposite signs and why they cancel out eachother?
If you applied a sine wave voltage across an inductor you would see that the current lags the voltage by 90 degrees. If you did the same with a capacitor, the current would lead by 90 degrees. 

The picture below is a particular example of where the inductor and capacitor currents have the same magnitude (electrical resonance) and, are opposite by 180 degrees (one leads by 90 degress and the other lags by 90 degrees): -



The values in the above example produce equal magnitude currents at 1kHz.

If you put them in parallel across the same 1 kHz supply, the currents would exactly cancel and there would be no current taken from the supply. This is the AC analysis and is called parallel resonance. Just in case you cannot see that, here is a picture that makes things a little easier on the eye: -


Hopefully, if you look at the upper and lower red sinewaves you should see that the two currents are exactly antiphase (180 degrees apart) and the net current is zero.



If you now put them in series they share a common current but, the voltage on the inductor will lead by 90 degrees whilst the voltage on the capacitor will lag by 90 degrees i.e. the two voltages are exactly 180 degrees out of phase.

When you have two series voltages that are exactly out of phase by 180 degrees the net voltage across the pair is zero. Can you see this?

It's the same with two batteries - if you have two 9 volt batteries connected in series but one is the wrong way round, the net voltage produced is exactly zero volts.

If you could make a resistor that had a negative resistance of "R" and put it in series with a conventional resistance of "R" then, the total series impedance is R + (-R) = 0. If you put those two resistors in parallel you would get infinite resistance. You can check this easily by using the "product over sum" calculation for parallel impedances/resistances.