Simple syrup is a solution of sugar dissolved in water used to sweeten cocktails (to balance tart citrus or similar). Many older cocktail recipes call for regular sugar to be added, but it is difficult to make sugar dissolve in a cold cocktail, often resulting in a grainy texture or unincorporated sugar resting at the bottom of the glass, both of which are usually undesirable. Simple syrup incorporates much more easily, and thus is a useful ingredient to have on hand.
This can be made in many ways, but is most often made by heating a 1:1 ratio of standard refined sugar and water until it begins to simmer, then removing it from heat and continuing to stir until all the sugar has dissolved. It is also possible to do the same with raw sugars such as demerara, which will result in a much more rich, caramel-like solution. Also, if one continues to heat a simple syrup solution, the sugars will break down into component sugars, resulting in a different flavour. Feel free to experiment with varieties and ratios to find the formula appropriate for your tastes and what cocktails you are making.
Regardless, it is an important ingredient to have on hand, as many cocktails call for it, and it can help greatly in rounding out and bringing together flavours.
Rock candy syrup and gomme are sugar syrups like simple syrup, though they are supersatured and thus have a much higher sugar content. Additionally, gomme contains gum arabic to help keep the sugar in solution. In practice, the three syrups can be used interchangeably with proper adjustment to the drink recipe.