There are 3 different strainers: the Hawthorne, the Julep, and a fine mesh strainer. 

The Hawthorne is the most common strainer you'll see behind the bar. It fits over a variety of different mixing glasses, and is typically used with a Boston shaker. It has a distinct metal coil, which traps pulp and ice. This strainer can be somewhat difficult to clean, but the coil can be removed for cleaning.  

The Julep is a classic strainer, and you'll see it in a lot of cocktail bars. It somewhat resembles a large spoon, and it has small holes to allow liquid to pass through. This strainer will be inserted into the glass slightly at an angle as you pour. It strains a bit slower than the Hawthorne, but it's far easier to clean.

You'll use the fine mesh strainer to finish off a cocktail, rather than use it alone. This is called a double strain; you'll use either the Hawthorne or the Julep over your mixing glass, while holding the fine mesh over your cocktail glass. This is used for cocktails where you want to ensure clarity of the drink.