Does kosher salt has different structure than regular table salt or it is issue of crystals size? Why some recipies want us to use one instead another?
There is no chemical difference between table salt and kosher salt - both are sodium chloride. Kosher salt just has bigger crystals. However, some table salt brands are iodized, meaning that iodide salts have been added to help us maintain a healthy dietary intake of the essential nutrient iodine. Kosher salt is never iodized.
Sea salt is a different matter. Since sea salt is obtained from sea water, it may have other minerals in it, which perhaps alters the taste.
Kosher salt is used in some recipes as a matter of taste - the bigger salt granules take longer do dissolve, and so there may still be little pockets of salty crunchiness. Kosher salt is also used when this property is necessary, for example in the curing of some meats. For more on the differences in culinary use between the various types of salt, you should consult Seasoned Advice.