This is the unit used for weighing stones and is how we refer to their size. All stones vary in density so an emerald, and a diamond, for instance, of the same carat weight would be different sizes. 1 carat is equal to 0.2 grams. 

Mohs Scale
The measure of how hard a stone is. Ranging from 1-10 with diamond being the hardest at 10. Softer stones are prone to scratches and may chip with a good knock, they also are much more likely to break whilst being set into the piece. Its good to consider the hardness of your with how you're going to wear the piece. To wear a ring everyday, depending on your lifestyle, you might want a hard wearing stone. Or to be prepared to have a more fragile stone replaced if it does chip. 

How clear or see through a stone is. The better the clarity the more light is let in so the stone will have more sparkle. Very low clarity stones can be nearly opaque. Emerald for example often has a low clarity and can appear cloudy.

In mineralogy an inclusion is any material that is trapped inside a mineral during its formation. They occur more is some stones than others and effect the clarity and value of the stone. Contrary as we are, we're a fan of the odd inclusion. You can see how the crystal was formed and it looks organic, interesting, not flawless. 
The rarer the stone, the more expensive it will be (apart from diamonds). When pricing stones jewellers and stone dealers will often talk about the four C's. These are; carat, colour, clarity and cut. As a general rule, the more expensive stones will be larger, brighter/stronger in colour, clearer/most flawless and the best cut. But money don't buy style darling, we think that taste is more important than grade, as long as you love it, that's all that matters. 
The modern birthstones referred to have no historical origin. The list was first published by Tiffany's and Co in 1870 and has been added to and switched around by various jewellery associations ever since. It has been described as "nothing but a piece of unfounded salesmanship." So if you don't like your birthstone, don't worry, just pick a different one.

Most of this information has been gathered from endless conversations with various stone dealers and gemologists along with our own opinions.
The rest has been fact checked on wikipedia.