It's a normal effect. If you play a house mp3 on a normal speaker turned loud, you could feel the air movement. The speaker moves back and forth on a certain frequency for each sound (the speaker is made from a membrane a magnet and some coil for magnetic field generator....).
The subwoofer is for low-frequency so it moves slower, moving higher mass of air.
Sound *is* vibrations in the air. Your hearing is air causing your inner ear to vibrate.
Speakers work by causing a cone inside to vibrate at certain frequencies (which our ears interpret as "pitch"). This is done using magnets which will oscillate in the presence of a changing magnetic field which is caused by the electrical current (also oscillating) being provided to the speakers.
You notice bass sounds moving the air because they are of lower frequency which means you can detect this with something less sensitive than your ear - your skin. Higher frequencies are much harder to notice.
On top of this, lower frequency (bass) sounds tend to be in the range which will cause resonance in certain parts of your body, causing these parts of your body to vibrate themselves on certain notes, making them much more noticable.