Both these words are uncountable nouns: Do you have a lot of luggage? Two pieces of luggage have gone missing.
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Suitcases are the same as luggage, but you cannot say two luggages as luggage is a mass noun. When you travel you have luggage, or two pieces of luggage. Mass nouns use measure words like pieces of to make plurals.
Common nouns are everywhere. Look around you now and you will see many common nouns: door, window, tree, sky, cat, truck, road, computer, phone, bottle, ball, luggage, people, hair.