What are Atmospheric Rossby Waves and how do they affect the weather?
Where to start....
If you take a look at midlatitude weather on any given day, you will find a pattern of upper tropospheric winds that look like this (source: NWS):
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Notice how it looks like a wave? These are called Rossby waves. You may imagine that the jet stream is imbedded in the ups and downs of these pressure contours. If we take a look at a typical winter storm (here is a random storm from 2003): 
Notice how wavy the 500 mb heights are? Don't they resemble the first figure?
Now you might say &quot;I've seen those things and they definitely move west to east.&quot; This is true. That is because the prevailing wind is westerly (in the above examples). This can be shown in a mathematical analysis, though it should be noted that baroclinic Rossby waves (mostly in the midlatitudes) propagate slower than barotropic Rossby waves, which is what this analysis works on.
I can probably into more extensive detail about Rossby waves, but I'll just end here.