Learn Russian: У Ивана есть велосипед.

In Russian, to say someone 'has' something, we use the preposition У plus the Genitive case. Because Иван is a masculine name ending in a consonant, we add -а to make it Genitive: У Ивана (literally 'By Ivan').

So the sentence literally translates to 'By Ivan there is a bicycle'?

Exactly! Russian doesn't usually use a verb for 'to have.' Instead, possession is expressed by saying that something exists near or by someone. У means 'by' or 'at,' and есть means 'is' or 'exists.'

What case is the word велосипед in?

It is in the Nominative case, which is the basic dictionary form. Even though 'bicycle' is the object in the English translation, in Russian grammar, the bicycle is the subject of the sentence (literally, 'A bicycle exists by Ivan').

Could I leave out the word есть in this sentence?

Not if you just want to say he has a bicycle. You need есть when the main point is the fact that he possesses the object. You usually only drop есть when describing a specific feature of something we already know he has, like 'Ivan has a red bicycle'.