"Are table salt, sea salt and kosher salt interchangeable? How do I know when to use each?"
Common table salt is mined salt that has been iodized (to prevent thyroid disease) and treated with anti-clumping additives. Sea salt and kosher salt are generally not treated with additives. Table salt is commonly sold finely ground; sea salt is sold in both coarse and fine grains, and kosher salt is coarse.
As a cook, you need to know that fine and course salt measure differently. Most recipes, unless they specify otherwise, assume you are using table salt. You may substitute fine sea salt in the same amount, but kosher salt, because it has larger grains, measures differently. You will need about 1 1/8 – 1 1/4 teaspoons kosher salt to get the same amount of "saltiness" is 1 teaspoon of fine salt.
The best way to understand the difference in flavor is to taste table salt side-by-side with sea and/or kosher salt. Table salt has a sharper, more chemical flavor, which sea and kosher salt taste, well– purely of salt.
For more information on using salt in recipes, see 7 Ways with Salt.