It's just like У меня есть which would translate to "By me there is" except the subject is different. I'm studying this in college, and have my textbook handy (Troika, 2nd Edition by Marita Nummikoski), I'll check with my professor, but I don't think the У is strictly necessary. (By which I mean dropping it in conversational speech would not be considered colloquially wrong, however to be properly written, the У is required)
When someone possess/have something, you say phrase like У (possesor in genetive case - родительный падеж) //меня/тебя/мамы/кота// есть #something/someone in nominative case. У кота есть лапы. (A/the cat has paws) У меня есть машина (I have a car). Sometimes the word есть is dropped, esspecially if the object of possesion has adjective. У меня большой дом. У тебя/вас хорошие друзья. (I have a big house, you have good friends. If you want to say, that someone doesn't have something/someone, you must replace the word есть with нет and also put the object in genetive case. У меня нет (большого) дома. У тебя/вас нет (хороших) друзей. У мамы нет сестры. (I do not have a (big) house. You don't have (good) friends. Mom doesn't have a sister.
English prefers to express ownership and “possession” with the verb “have”. In Russian “existence” is almost universally used instead (in the official/academic style «иметь» to have is OK to use).
Use it like that:
У A есть X ~ by A there is an X → A has an X
The owner is in the Genitive case (more on that later) while X is formally the subject. For now we will only study the Genitive form for some pronouns.
So the possessor of the object (the person directly after the "у") will always be in its genitive spelling regardless of if we're saying that the object exists or not (есть/нет)? And the object itself changes to genitive spelling only if were saying it doesnt exist (нет)? Is this correct?
"Мамы есть сестра" would literally mean "moms are sister" which makes no sense.
"У" is a preposition. Preposition never have an exact match between different languages, but in this context it's usually understood as "by" or "at the possession of". "У мамы есть сестра" literally translates as something like "by mom there is a sister" or "in mom's possession there is a sister".