Unfortunately, the prerequisites are a little too extensive to mention most aspects here, and your familiarity with each depends with your previous study. Ideally, that's what high school and college degrees are for. Thus, I mention concepts particularly relevant for our mission. If you do not understand these, you won't understand parts of the rest. If you have questions, please send me a message. If any of your inquiries are more than I can answer, I'll attempt suggesting the best reference.
You can study geometry for years, ranging between elementary school to doctoral study and beyond. It is quite relevant because meteorology greatly involves descriptions of weather systems & variables in 3-dimensional space, most often depicted on 2-dimensional maps. Though you may understand these topics well, you should be aware that I strictly refer to them as they are defined. I.e., if I write line, I mean exactly that - not curve or arc.
I do not prove nor derive anything, nor discuss topics with much detail here. Among the more useful WWW resources for this are the following sites : Elementary Geometry Tutorial & Geometry Formulas and Facts, though they also are not comprehensive - a geometry textbook ideally being required. Even so, proofs & derivations are little help for our purposes - more so are calculations using concepts important to us (which are proven or derived).
Below are concepts I'll most often use :
Point, ray, line, curve, parallel, perpendicular, angle, right angle :
Plane, triangle, right triangle, trigonometric functions :
Adjacent, opposite, symmetry, axis (wrt means "with respect to") :
Center, circle, circumference, radius, diameter, arc, curvature :
Ellipse, ellipsoid (not shown) :
Sphere, and great circle arc, angle, & triangle :
Some equations in diagrams refer to coordinate systems described below.
Among most relevant applications are :
Plane : With no elevation change, ground can locally be considered as approximately a
horizontal plane :
Though it contains a slight curvature, such is of little consequence regarding most weather
phenomena (but very significant regarding large scale weather systems such as Highs and Lows
seen on a weather map). Mountain slopes can be approximated as planes tilted various angles
wrt horizontal.
Trigonometric Relations : Many are relevant, but most so are sine functions and the
Pythagorean Theorem. A consequence of these is that perpendicular components of a right triangle
are greater than their opposite angles indicate, greatest difference for 45° angle.
Circle : If you consider regular polygons (figures with each side equal, such as a 'perfect square, hexagon', etc.), increasing the number of sides to infinity produces a circle. Each place on a circle is same distance from its center.
Sphere : Rotating a circle in any direction except in its plane produces a sphere.
Ellipse : An ellipse is symmetric wrt 2 axes. A circle is an ellipse with eccentricity
0. Planet and Moon orbits are very nearly elliptical. Kepler's Laws describe the
basic
character of orbits, though the
details can become much more complicated 
.
Earth's orbit is very nearly circular, with eccentricity presently .0167.
Ellipsoid : An ellipsoid is an ellipse rotated around one of its axes. Earth is nearly an ellipsoid, which is often an adequate approximation for basic astronomical calculations. Properly, Earth is an oblate spheroid.
Great Circles : Great circle arc, angle, and triangle refer to those on a spherical surface. This is relevant because distance between any 2 points on earth can be estimated using relations shown if their angular locations are known (i.e., latitude and longitude). Assuming Earth is a sphere is fine for many meteorological calculations, introducing less error than other factors involved.
These are abstract ways of defining locations of objects. Notice that I mentioned geometry first, avoiding reference to these as much as possible; because curves, shapes, angles, distances, etc. are most relevant, rather than position specification. Such is necessary though. I mention 3 systems most relevant :
Cartesian Coordinates : These are defined according to perpendicular axes, the
abscissa and ordinate, typically labeled X & Y, respectively :
Location of objects are specified similarly as the point's shown. Equations previously shown
represent such objects on Cartesian coordinate axes. Such is often used as reference for
meteorological data - U & V analogous with X & Y, representing west-to-east and south-to-north
directions. I.e., a wind with positive U & V components is from a direction generally W & S.
Polar Coordinates : These are defined according to distance from the origin of
coordinate axes, and arc from a reference ray :
Radar images and hodographs are among common meteorological uses of polar coordinates - storms'
distance and direction from a base radar are specified.
Spherical Coordinates : These are often used for earth, being a good approximation
of a sphere. They are defined according to distance from origin and (spherical) longitude and
latitude angles :
If you imagine a polar coordinate system flat (as on a table), longitude is polar arc angle
(from reference ray) and latitude is elevation angle above horizontal (table), as shown. Thus
for Earth, the equator is in the polar coordinate plane, its North and South Poles along the
axis perpendicular to that plane. Because Earth is not spherical, radial distance is not same
everywhere on its surface, but is nearly so.
Distance Between 2 Points on Earth : Using the formulas for Great Circle Triangles above, distance between any 2 points on a sphere's surface can be simply estimated :
D = (6367.5)(3.14159/180) cos-1((sin(40.8°)sin(53.6°) + cos(40.8°)cos(53.6°)cos(73.9°-113.6°)) = 3262.8 km
As mentioned above, Earth is not a sphere, but an oblate spheroid :
though differences are not great.
Wind Direction : Among the most commonly misinterpreted meteorological parameters
is wind direction. People don't often have trouble understanding that wind speed is 32 mph
(especially if it's blowing in their face), but if you ask them if such a wind is called a
northeast or southwest wind, many are unsure. I'm gonna seemingly add more confusion - define
winds differently for each hemisphere ! Because weather systems tend to circulate W to E around
the poles, placing the pole on top of a map for either hemisphere is a natural representation.
This idea became evident for me while forecasting for the World Solar Challenge in Australia.
If I turned Australian weather maps up-side-down :
things were exactly as in
the Northern Hemisphere, except that circulating from W to E meant right to left instead of left
to right. The days when all analyses are accurately portrayed on 3-D images which can be viewed
from any perspective are likely long from now, so this idea is currently relevant. I won't change
directions we are familiar with (i.e., the North Magnetic Pole's location remains so anywhere on
the globe), but headings. I understand that such is perhaps not recommended for many other
applications, mainly because it requires a reversal from one side of the Equator to the other
(which programming in atmospheric models is not extremely difficult). As can be seen, using such
specification allows similar meteorological significance for similar wind headings.
My suggested convention is irrelevant regarding the following definitions :
E.g., a wind called W (west) @ 16 knots blows from W to E (east) with speed 16 knots. Such is also commonly called a westerly wind, which blows (to the) east. Modified base map above was obtained from Tiger Mapping Service. I apologize if this is obvious, but I've seen many misinterpretations. Note that wind velocity means (both) direction and speed. Quite often, the term velocity is used when speed (only) is meant.
Storm Movement : This typically means direction storms move to or toward. Consider for example, a hurricane warning and strike probability map :
Next I plan to briefly finish discussion of prerequisites, then begin a brief discussion of
topics listed.
Text is copyright of Joseph Bartlo, though may be used with proper crediting.