Winds and convection currents keep moving air and changing the density in ripples like the ripples in a stream. Simetimes you can see the ripples in cloud patterns.

Those ripples behave like moving lenses between you and the stars. They move the direction of the light so that the star seems to change its place, to "wobble", and so that at one moment more light reaches you from a particular star, and a moment later less light. The effect is to make the star seem to flicker more and less brightly, and shimmer as it moves about a little bit.

Sorry!!! Not again!

The word you're looking for is twinkle :)

If you were outside the Earth's atmosphere then you would see the variability in brightness reliably. Looking through the atmosphere you will also see flickering caused by the bending effects of air. For example, Sirius is the sky's brightest star and twinkles like mad when viewed from here (Ireland) but the main reason is that it reaches only a low elevation so we see it through loads of air.

Variable stars (all stars are variable at some stage in their life-cycle) change in brightness over time-scales varying from seconds to many years. The magnitude of the change can be enormous and it's worth having a read about Cepheids, Pulsating variables, Eclipsing variables and more... but I suspect your question relates more to the atmospheric effect mentioned above.

EDIT: cross-posted with Jon there. Sorry for the repetition.

The atmospheric explanation of twinkling has been given for a century or more, and as far as I can see, it does not explain the phenomema.

Wikipedia says that (a) Sirius (one of the biggest closest stars) subtends 7.56 seconds of arc, and (b) the human eye has an acuity of around 0.35 minutes of arc, which is 21 second of arc. So a stream of photons from Sirius without atmospheric or visual effects is more likely to fall down the cracks as to hit one single visual receptor.

The same constraint affects even extra-terrestial telescopes like Hubble. Whatever their magnification, no telescope should be able to resolve a remote star into a disk as you see on photographs. The spreading of the disk is an artefact of the detection mechanism, not a proper image of the original source.

The eye naturally make tiny involuntary movements (microsaccades) to spread the visual load over adjacent receptors. I believe this constant adjustment accounts for the perceived varying brightness of stars in direct vision.