Can I become a permanent resident through my Mom?

Your mom will have to petition for you once she is a permanent resident and it will take a minimum of 2-3 years if you are still 21 at that point; 4-6 years if you turn 21 before a visa number becomes available.

Your mom can get her green card as an immediate relative through your brother when he becomes a citizen. But there is no derivative green card for you. Once your mother becomes a PR, she can petition for you. She should do so right away to see if you can come before you turn 21. Your brother can also petition for you. But the wait time is long.

Your brother cannot petition for your mother until he is a US citizen. If he gets his green card through marriage to a US citizen, he can apply for naturalization after he has been a legal permanent resident for at least 2 years and 9 months.

Your brother may also file for you when he becomes a US citizen. However, if you review the Department of State Visa Bulletin you will see that there will be a substantial wait time. Your mother may file on your behalf once she has status (she will be an immediate relative and not have to wait for a priority date once your brother is a US citizen). That will also take a substantial number of years however. Until then you may want to explore student visa options.