The lapse rate is basically the rate of decrease in temperature with height. Shallow lapse rates mean that the enviromental temperature is cooling quite slowly with height. Steep lapse rates mean the environmental temperature is cooling rapidly with height. Here's a rough example I've done.
You can where I've marked the shallow lapse rate, I've marked where it starts around 12,000ft. The temperature there is roughly 4 - 5 degrees C. At 18,000ft where I've plotted the other line, the temperature is around -3C. So over 6000ft or so, the temperature has cooled around 7 degrees. So the lapse rate is reasonably shallow.
The steep lapse rate starts around 18,000ft. The temperature there is around -4C. It ends at around 22,000ft. The temperature there is around -16C. So over 6,000ft the temperature has cooled around 12 degrees. The lapse rate for that portion of the atmosphere is comparatively steeper.
That example is only to show lapse rates for portions of the atmosphere. Just shows the difference between shallow and steep lapse rates.
Here are some helpful links, for lapse rates of the atmosphere, adiabatic change, cooling, relation to thunderstorm formation, latent heat, ect. http://rst.gsfc.nasa.gov/Sect14/Sect14_1b.htmlhttp://www.youtube.com/watch?v=cq0zIZTnYpUhttp://geography.about.com/od/physicalg ... serate.htmhttp://www.bom.gov.au/aviation/data/abo ... gram-c.rtfhttp://eesc.columbia.edu/courses/ees/cl ... _phys.htmlhttp://www.youtube.com/watch?v=zJCq9eqI ... re=relatedhttp://www.youtube.com/watch?v=XH_M4jIt ... re=relatedhttp://www.theweatherprediction.com/habyhints/30/http://en.wikipedia.org/wiki/Atmospheric_convectionhttp://science.jrank.org/pages/6828/Thu ... pment.html