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Yes, an inductor is the opposite of a capacitor in some ways. If you put current into a capacitor, it builds up a voltage. If you apply a voltage to an inductor, it builds up a current. If you short a charged capacitor, the result is a high spike of current. If you open a charged inductor, the result is a high spike of voltage.
In addition to the behaviors they exhibit alone, they also are opposite in how they react in a circuit. Capacitors tend to pass AC current and reject DC current, where an inductor tends to reject AC current and pass DC. This is why we use them together in filter arrangements.
Another aspect of how they affect a circuit in opposite ways is the phase shifts they create. Current leads voltage by 90 degrees in a purely capacitive circuit, but current lags voltage by 90 degrees in a purely inductive circuit. In a purely resistive circuit, voltage and current are at the same phase.
Mathematically speaking, inductors and capacitors are "inverses" of eachother. You can use a capacitor to negate the effects of a inductor (in controls speak, you can use a pole to cancel out a zero).
Note: Xc = 1/(j*w*c) and Xl = j*w*L. If you set them equal and solve, you see that they are by definition Mathematical inverses at w = 1/sqrt(LC), a more familiar term would be called 'resonance.'
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