I've been researching Atmospheric Rossby waves but from the multiple sources I used they all seem to be in conflict. Some say Rossby waves are parcels of air moving from east to west due to the coriolis effect (which seems like a description for a jet stream) while others sources say it's the curving of the jet streams. I have a hard time understanding Atmospheric Rossby waves and how they affect the weather and climate.
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):
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 "I've seen those things and they definitely move west to east." 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.
Maybe you can try to understand Rossby wave by appreciating the analogy between (internal) gravity wave and Rossby wave.
Two distinct features of geofluid are stratification and (differential) rotation. Stratification, or vertical variation of density, produces internal gravity wave. Similarly, differential rotation (or variation of rotation), gives rise to (barotropic) Rossby wave. Their difference can not be easily identified by observing the tracer (temperature, density etc.) field in the figure below, without knowing the coordinates.
Vertical decreasing density with altitude gives a fluid parcel a restore force when it is displaced upward/downward; Likewise, meridionally increasing rotation also gives a restore force when a parcel is displaced northward/southward. Therefore, a fluid parcel cannot move freely in the vertical and meridional directions due to these restore forces (you probably already notice that the contours of many things on rotating planets are horizontal lines or zonal lines). Such restoring forces actually generates respectively the internal gravity wave and Rossby wave. The gradient of potential temperature (density of mass), as well as the gradient of potential vorticity (density of rotation) here is a measurement of the restoring force.
The internal gravity wave can bring lighter (heavier) fluid downward (upward). Similarly, the Rossby wave brings colder (warmer) fluid equatorward (poleward) and thus affects the weather. In addition, the Rossby wave also squeezes the atmosphere so that the water vapor condenses into rainfall, affecting the weather.