Learn Russian: Ваш счёт, пожалуйста.

While the waiter could use an imperative verb and say Возьмите ваш счёт, пожалуйста (Take your bill, please), Russians often drop the verb entirely when handing something to you. Just naming the object and adding пожалуйста is a very common and perfectly polite way to offer something.

Who is normally saying this sentence, the customer or the waiter?

This is what the waiter says when handing the check to you. If you are the customer asking for the check, you wouldn't say "your bill." You would simply say Счёт, пожалуйста (The bill, please) or Можно счёт? (May I have the bill?).

Why is it ваш instead of ваша or ваше?

Ваш is the formal version of "your" and it must match the gender of the noun it describes. Because счёт ends in a consonant (т), it is a masculine noun, so we use the masculine form ваш.

I see letters in пожалуйста that I don't really hear when native speakers say it. Are they silent?

Yes! The letter й and the letter у are almost always skipped in everyday, natural speech. Instead of sounding out every letter, it is usually pronounced like "pa-ZHAL-sta".