Learn Russian: У меня есть виза.

Russian doesn't use a direct verb like "to have." Instead, possession is expressed as "At me there is...". The word У means "at" or "by." After У, you must use a specific grammatical form (the genitive case), which changes я (I) into меня.

What is the exact role of the word есть in this sentence?

есть means "is" or "exists." Since the sentence doesn't use a verb for "have," есть provides the action. The whole phrase У меня есть виза literally translates to "At me exists a visa."

If I "have" the visa, shouldn't виза change its ending because it's the object?

In English, "visa" is the object. But in Russian, because the sentence literally means "At me there is a visa," the visa is actually the subject (the thing that exists). Therefore, виза stays in its basic dictionary form (the nominative case) and doesn't change.

How should I pronounce У меня?

In Russian, short prepositions like У blend into the word that follows them. You should pronounce it as if it were a single word ("oo-mee-NYA"), without any pause between У and меня.