A carat is a weight, not a size
The metric carat has been exactly 0.2 grams since 1907, and it divides into 100 points — so a 0.25 ct stone is "twenty-five points". Because it measures weight, and weight grows with the cube of the dimensions, carat weight and visible size are not proportional. Doubling the weight of a round brilliant from one carat to two increases its width by the cube root of two, which is about 26%. A one carat round is roughly 6.4 mm across and a two carat round roughly 8.1 mm.
This is the single most useful thing to understand before comparing two quotes, because price rises far faster than size does. The step from 0.9 ct to 1.0 ct costs a great deal and adds about a third of a millimetre.
Shape changes how big it looks
Face-up area — how much of the stone you actually see looking down at it — does not rank the same way as carat weight. Elongated, comparatively shallow shapes spend more of their weight on the outline: marquise shows the largest face-up area for a given weight, followed by pear and oval. Princess, cushion and asscher are cut deep, so more of the weight sits below the girdle where nobody sees it, and they read smaller than their weight suggests.
There is a real trade-off available here. Fancy shapes generally cost less per carat than rounds, because cutting a round wastes more of the rough. So a marquise or oval can be both larger to look at and cheaper than a round of the same weight.
Estimates from carat weight assume a normally cut stone. Two one-carat rounds can differ by several tenths of a millimetre face-up depending on how deep they are cut, which is exactly why two stones of identical weight can look like different sizes.