Ask a bartender why the ice matters and you will hear "because it's half the drink". Ice does two jobs at once: it takes heat out of the liquid, and as it melts it puts water into it. Both jobs happen at the surface of the ice, so the one thing that decides how a drink turns out is how much surface you give the liquid to work with.
Melting ice soaks up a lot of heat: about 334 joules for every gram that melts. That is why ice is such a good chiller, and why you cannot get cold without some water. The water is not a flaw. A stirred drink is usually taken to around 15–20% water by volume and a shaken one to around 20–25%; the water softens the alcohol, opens up flavour and takes the burn off. The only question is how fast you get there, and how much more you get if the drink sits.
Ice straight from a home freezer, at around −18 °C, is colder than melting ice. It can absorb roughly 38 joules per gram just warming up to 0 °C before any melting starts, around 11% on top of the melting heat. That is why fresh-from-the-freezer ice chills a little harder and dilutes a little less than ice that has been sitting and sweating.
Take a cube with sides of length s. Its volume is s³ and its surface is 6s², so surface per unit of ice is 6/s. Halve the cube's side and you get the same ice with twice the surface. A glass full of crushed ice has many times the contact area of one big cube, so it pulls heat out of the drink much faster.
That has a second effect. If every face of a cube melts inward at about the same pace, a cube lasts for a time proportional to its side. A 50 mm cube lasts about twice as long as a 25 mm one, and about eight times as long as crushed pieces of 6 mm. Real melting is messier (stirring, glass shape and room temperature all matter), but the trend holds, and it is the reason bars treat ice size as a choice rather than an accident.
In the mixing glass or shaker, use plenty of standard cubes. Here you want the drink chilled and diluted in under half a minute, and then you strain the ice away. Small surface area is the enemy: a stirred drink with one large cube will still be warm and strong after a minute of stirring.
In the serving glass, think bigger. Once the drink is poured over the ice, you want it to stay cold without turning to water. A single large cube or a few big ones keep the surface small, so the drink dilutes slowly as you sip. Using fresh ice for serving, not the ice you just shook with, helps too: that ice is already wet and partly melted.
Crushed ice is its own thing. A julep, a tiki drink or a mocktail with lots of fresh mint is meant to be cold and a little watery, and the ice usually vanishes while you drink. That suits a hot day and a long drink, and it works as well for an alcohol-free spritz as for a cocktail.
Cloudy ice has tiny trapped air bubbles and dissolved gas. Air makes ice less dense and weaker, so cloudy ice cracks more easily and melts a little faster than clear ice of the same size. Clear ice is made by freezing water slowly from one direction so the air is pushed out ahead of the freezing front. A common home method is to put an insulated cooler, lid off, into the freezer and freeze water from the top down, then cut the clear part from the block. It looks good, and it is slightly better ice, but size matters far more than clarity.
As always, enjoy drinks in moderation. Water is the other half of the picture: it's worth a glass alongside any cocktail.