REGULAR SUGAR is made up of glucose and fructose molecules which are linked together and like to re-connect. Stray grains of undissolved sugar can trigger a cascade effect of reconnecting the molecules which causes your sugar to crystallize.
INVERT SUGARS are made up of glucose and fructose molecules that are not linked to start with and therefore do not try to re-connect. Adding some invert sugar can discourage crystallization which helps with a smoother caramel and a more velvety texture to ice creams and sorbets.
Glucose an invert sugar which often comes from corn but can also come from potato or wheat. The consistency is thicker than corn syrup or sugar syrup (made from equal parts sugar and water). This liquid is 100% glucose, with a thicker viscosity and because it is less sweet with a neutral taste, I prefer this in most cases when I need to add some invert sugar to my recipes (sorbets, mirrorglaze, sugar shards) when I'm looking for a smoother outcome or want to avoid crystallization when working with sugar.
Corn Syrup is also an invert sugar, but as the name states, is only made from corn and has had some of the glucose molecules converted to fructose. This glucose and fructose liquid is runnier than liquid glucose. Corn syrup is easier to find in supermarkets has a higher water content and is slightly more sweet than liquid glucose. It can be substituted for glucose but you need to adjust your recipe for the extra water content in corn syrup.
Sugar is Sucrose, is also made up of glucose and fructose molecules, but unlike invert sugars - the glucose and fructose molecules are like lovers on a mission to reattach to one another! When they do re-attach themselves, this is what causes crystallization. A 'Simple Syrup' is a sweet liquid made by simmering equal parts sugar and water into a syrup. It tastes the best out of the three liquids, but can also turn white and grainy if crystals form.
Does crystallization matter?
It all depends on what you are making! Crystallization doesn't matter in many recipes such as cakes and cookies, but for sorbets and caramels, I like to add some invert sugar to achieve a smoother texture. Acid such as lemon juice can also discourage crystallization, but because you are adding lemon, it may or may not be suitable for your recipe.
What about natural sugars?
If you have a thing for natural sugars, like cane sugar, honey, agave and maple syrup, be wary when substituting as they can impart a strong flavour. Unless I am after a specific flavour profile, sugar is sugar and too much of a good thing is bad for you regardless of whether it is processed or natural. If you feel you would prefer to use a natural sugar, research its properties and run some tests!
Don't give up on your caramel too soon!
A great tip I've discovered is that prolonged HEAT can INVERT some of the sugar and even reverse it's effect. So when melting sugar, if it begins to go white and crystallized, try lowering the heat and continue cooking the sugar as it will often re-melt into a caramel with a smooth consistency.