Overall, the most noticeable implement used in forecasting is the weather map. Meteorologists can see how winds will blow systems around (advection) and change their shape (deformation). If you were to visit the link below, you would learn that wind flows clockwise around a high pressure system (in the northern hemisphere) and why, as well as how atmospheric pressure relates to observable conditions (sunny, rainy etc.) And why.
You have heard of the troposphere, stratosphere, mesosphere and thermosphere above us. They are all very different from each other, but consistent over the entire planet. Some maps show important large scale features such as air masses and jet streams. In order to be accurate, they have to incorporate influences such as the curvature and rotation of the earth.
Again the presence or absence of moisture is crucial in all aspects of forecasting. For instance, we use that information for storm prediction and many more complicated things. That is because the latent heat stored in water vapor serves as a potent energy source to drive mechanical processes, which can then extract more heat from the vapor and so on. Another construct, vorticity, further enhances the twisting and lifting power of the atmosphere.
The waves and streamlines on an assortment of weather maps carry special meanings for meteorologists as well. Again, an intuitive knowledge of physics greatly helps the forecaster understand, explain and prognosticate the situation.
About the Author
A word or two about myself
I sometimes go by the nickname weather man, or even piano man. I have a B. Sc. in physics and meteorology - University of Alberta in Edmonton, Canada. Much of my work has been in the field of atmospheric modeling. I also have been a musician of one kind or another all my life.