Crown- The Crown is what most might call the "top half" of the stone. This is the part of the stone you see when looking at a finished piece of jewelry. It includes everything from the girdle up.
Pavilion- The pavillion is what you might call the "bottom half" of the stone. This is the "pointy end" of the stone, and includes everything from the girdle down to the culet. This part of the stone acts like a reflector and is probably the most important part of the stone when it comes to the overall brilliance of the gem.
Girdle- The Girdle is the thin outer edge of the stone where the crown and pavillion meet. This is where the Jeweler puts the prongs when setting the stone.
Culet- This term is usually used to describe a small flat facet cut on the bottom of the pavillion on the "point." It was origionally used to help prevent chips on the tips of diamonds, but is rarely used anymore. I list the term here, because you might see refrence to the "culet" of a stone in refrence to the point on the pavillion, when there is no flat spot. This is one of those terms that seems to be used regularly, even though its not exactly the "correct" term.
Table- This is the large flat facet on a stone on the top of the crown. This is what you see when you look at the top surface of a gemstone when it is set. The table is important in the optical performance of the gemstone, as it helps allow light to enter and leave the stone. Not all designs have a table, but most stones that you will see do.
Facet- This refers to a flat polished area on a stone. The combination of many facets is what gives the finished stone its "sparkle" as the stone is moved, the light reflects off of the various facets on the stone.
Window- This term is used to describe when a stone is cut with angles on the pavillion that are too shallow. Light rays can pass through the bottom of the pavillion, and you can look right down through it like a window. In a properly cut stone you should not be able to look straight down through the bottom of the stone and see what is on the other side when you are holding the stone crown up. This is often also called a "bow-tie" on a marquise or oval shaped stone, because the window is often shaped like a bow-tie.
Meet- Or often called "meetpoint" is the junction where several facets all meet in one location. In a properly cut stone, every facet that meets should do so at exactly one point. In a stone that is not cut properly, sometimes one facet(or several) does not quite come to the same point as the rest, sometimes one facet(or several) are cut too far and go right past the meet. A stone where the meets are off just is not as pleasing to the eye as one where everything lines up properly.(Have you ever been in a room where the wallpaper didn't match at the seams and it didnt look too good? same thing here)
Critical Angle- This is the shallowest angle that a stone can be cut at without windowing. Think of it like looking at a swimming pool. From a distance, you will only see the reflection of the sky on the surface of the water because your viewing angle in relation to vertical is very steep. As you get closer to the pool so you are looking down into the pool more(with a shallower angle in relation to vertical) at a certain point you no longer see the reflection of the sky, but you can look right down through the surface of the water and see what is below. The polished facet on a gemstone acts the same way as the surface of the water. Different materials have different refractive indexes, and critical angles, so the angle a stone is cut at for best performance varies depending on the type of gemstone.
Refractive Index(RI)- I could go into a long boring scientific explanation of the refractive index of a gemstone, but I don't want to bore you too much. Bassically, the refractive index is a number that shows how much the gemstone will bend a ray of light. The higher the refractive index is, the more light is bent when it enters or leaves a gemstone. The higher the refractive index is, the shallower a stone can be cut, because the light will bounce off of a facet at a shallower angle. This is why if you use a design optimized for CZ(with a high RI) and cut it in quartz with a much lower RI, you will have a problem with windowing in the quartz stone.