A shaker comprises 2 parts. The first is the tumbler in which the liquid is actually mixed and the second is the top. Some shakers come with strainers which are built into the top. Shakers are usually of two types. The first being what is called a Boston shaker which has a glass bowl for visual display of the liquid (and which chills faster) and the second being a classic all metal one. Some shakers also come with a vinyl coating around the mixing tin to prevent your hands from warming the drink and to prevent scratches to the steel. A bit more expensive though. A mixing can is just the bottom half of a shaker. A 16 oz. size is the optimum one for the mixing glass.

Avoid serving ice with your hands (not hygienic) or with a glass (may chip)

For making sure that you're pouring the required quantities of liquid. Make sure you get a double sided one, with one end for a chota peg (30 ml) and the other for a burra one (60 ml). Stainless steel ones are good.

For cutting garnishes.

For cutting garnishes on. It should be heavy and laminated and at least 8 inches in diameter.

Useful for mixing drinks and can also double up as a muddler - something to crush particular ingredients - say mint leaves when you're making a Mojito.

For use when the cocktail shaker doesn't come with a built in one. Keeps the ice and other pulpy parts of fruit juices out of your drink.

I don't think we need to tell you what a bottle opener is used for.