Learn Russian: У дедушки большая борода.

In Russian, to say someone "has" something, we use the preposition у + the Genitive case to mean "by" or "at" that person. So у дедушки literally means "at grandfather".

Why isn't the word есть (is/has) used in this sentence?

When talking about physical characteristics, body parts, or inherent features like a beard, Russians typically leave out есть. You simply say "At grandfather big beard." You would only use есть here if someone asked if he had a beard and you wanted to strongly emphasize that yes, he does exist with a beard.

Since дедушка is a man, why does it decline like a feminine noun here?

Even though дедушка is masculine in meaning and takes masculine adjectives, it ends in -а. Nouns that end in -а or -я follow the feminine declension pattern, regardless of their actual gender. So it behaves just like a feminine noun when changing cases.

Wait, the normal Genitive ending for an -а noun is -ы. Why does дедушки end in -и?

This happens because of the 7-letter spelling rule. In Russian, you can never write the letter ы after к (or after г, х, ж, ч, ш, щ). Because of the к in дедушка, the ending changes from -ы to -и, making it дедушки.