Purpose: Mass produced gem settings come in standard sizes. If a gem won't fit in a standard setting, one needs to be custom made. This involves much more labor and therefore expense.
Bear this in mind when purchasing a gem, or selecting a cut for a gem. There are circumstances where the value of a stone makes it worth cutting for maximum yield, then later paying for a custom made setting. In other circumstances, the cost of a custom setting isn't justified by the value of the gemstone.

Sizes: A gem does not need to be the precise size of the setting. A skilled metal smith can modify a setting to accommodate a wide variety of gem sizes.
Any round gem can be put in a standard setting. On other shapes there are a variety of factors to be considered. As a general rule, gems under one carat can be within .1 mm of the setting size. Over one carat, a .2 mm variance can usually be accommodated and sometimes more.

Weights: The weights given are for diamonds cut to ideal proportions. Colored stones are usually cut with greater volume than diamonds, hence they will weigh more than the chart sizes.
The density of the material will also affect the weight. For example a 6.5 mm round diamond, sapphire and opal, all cut in the same proportions will all weigh different. The diamond, (with a specific gravity of 3.52,) will weigh 1 carat. A sapphire, (specific gravity 4,) will weigh 1.14 carats. The opal, (specific gravity 2.15) will only weigh .61 carats.
There are subtle variations to this list. For example, some list a .25 carat diamond as measuring 4.1 mm. The point is not to take these figures too literally, but to use them as a starting point for estimating weights.