Ice is one of the most important aspects of creating good cocktails (unless you like room temperature cocktails). What type of ice you use can turn your cocktail into an entirely different drink. So it's important to be aware of the different types of ice and how each one can affect your cocktail experience.
The standard ice cube. Roughly 1 inch by 1 inch. Ice Cubes are the most commonly used ice types by bartenders, and probably the closet to what you have at home. They can fit in any cocktail glass, and because they are large & thick, they melt slowly, which is ideal for cocktails where you don't want to lose the original flavor.
Crushed/Pebble/Shaved Ice works best in drinks that require dilution. Such as the Moscow Mule and in most Tiki drinks. Because of their smaller size, crushed ice will dilute the fastest.
Round Ice Cubes are popular in spirit-heavy drinks such as the Old Fashioned. They melt the slowest due to their low surface area to volume ratio. This will ensure that your drink gets cold without getting diluted. Usually this is the hardest ice type to make at home and requires you to use an ice sphere mold.