My sister is a BRAT!!!

Hopefully, your sister is a teenager, you mentioned younger, but I'm not quite sure.  As I am not a psychologist, I can only offer tips and advice I have read, here is a website link about a few causes for the rebellion and another website link about how to deal with rebellious teenagers.  Again, I only hope that your sister is a teenager:

 
Teenage Rebellion
This is How It Goes!

Teenage has become synonymous with the word rebellion. As soon as a person hears the word rebellion one immediately paints a picture of sulks and tantrums and unreasonable behavior. Instead of taking it at face value we need to delve into the reasons behind it. What makes a normal reasonable individual to act in such an unpredictable way.

We need to better understand adolescent developmental stages to help us not take teenage behavior as a personal attack on us. By becoming familiar with these stages, we will increase our competence in encouraging teens to establish their sense of identity.

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  Teens are preparing to separate or individuate from the family. They are in the process of developing their values.
   
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  Teenagers must initiate this separation and often rebellion gives them the energy to do this. A teenager challenges rules and values as a way of establishing his or her individuality. Adolescents cannot do this in a vacuum, but rather through conflict and confrontation
   
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  Adolescents may be rude or make fun of parents and other authority figures and not want to be with them. In a teenager’s mind, defiance expresses autonomy and says that he or she doesn’t need parents in and often serves as a test of parental caring.
   
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  Due to body changes, there can be confusion about whether teenagers really do want to grow up.
   
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  Hormonal changes cause mood swings marked by tearfulness, heightened sensitivity, sudden flare-ups, an increased need for physical activity and inappropriate laughter and giggling.
   
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  Teens begin to work out their relationships with their peers to find out how they fit in.
   
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  Teens start relating to the opposite sex in a different way than they did when they were younger (where there were once friendships, romantic relationships and/or deeply felt negative emotions may surface).
   
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  Teenagers have a heightened need for privacy. Experiencing privacy gives them a new sense of control and autonomy. They need privacy to test things out for themselves without parent input.
   
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  Teenagers may feel all-powerful and all-knowing at the same time that they experience fears of inadequacy and failure.
   
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  Teens still need an adult to relate to, but in a different way than they did when they were younger.
   

The trick is to form a strong bond with a teen so that he/she feels close enough to reveal all the problems and confusions to his/her parents.

Rules without relationship give birth to Rebellion. 

source:  http://www.boloji.com/teens/articles/00312.htm

Dealing with Teenagers
How To Deal with Rebellious Teenagers

Teenagers are at a critical stage in their lives, which is the transition from being a child to maturing into an adult. During this period of their lives they are going through vast hormonal changes and are experiencing pressure from their peers, the media as well as other external influences. At this significant stage in their lives they need their parents to become their friends rather than their parents or the high authority controlling their lifestyles. They need to feel help and support throughout this stage in their lives.

During this stage they are continuously seeking independence from their parents through all aspects in their lives. They are beginning to develop sexual desires and question the existence of god and spirituality in their lives. They desire to be treated as adults, and they are in constant search for their identities, finding the groups in which they belong and, more importantly, finding themselves.

read more here

I am sorry to hear that you are having problems in your family.

And you can talk to your mom about it.

 In talking to your mom, you can let her know how uncomfortable you feel that she is being disrespected.  You should also let her know that as an older sister that it isn't your role to correct your sister as a mother would, you are their for guidance only.  Ask (not tell) what your mother plans to do to help your sister not treat your mom this way.  Tell her you are worried that if it continues, there is the danger that this attitude will rub off on the whole family, and you are scared of that.  The truth usually works.

If it is nothing or nothing changes, and if it continues, you will leave the room so your mother can handle the issue alone.  But only tell your mom this if she doesn't have a plan or you don't see her following through.

You can't do anything to your sister.  You are the older sister, and are expected to be a role model for her.  You can also give her advice, but you aren't her parent.

Also, try not to judge your sister too harshly: she might be going through a trying time, your family and yourself might not shown what correct social boundries are, she might be diabetic or hypoglycimic, etc.  But it's your mom who should be looking into it. 

If you get in the middle of this, you will have your sister's and mother's problems placed on your shoulders.  Not only will you continue to be angry with your sister, but you will begin to get angry with your mother as well for not being a mother.  For many young people, if they have respect for the parent, but are losing this respect, they will begin to lash out at the world...starting with the people closest to them, such as your sister.

If your mom has no idea what to do, it's ok.  She has lots of options, such as talking to parents, community groups, heck even yedda! 

Your sister is obviously angry about something concerning your mother and because she hasn't addressed the issue with her, is taking it out on her in other ways.  I'm assuming that she is not this way with you or other people.  When people try to "stuff" their anger, it is like trying to hold a beach ball under water.  Eventually it is going to come out and it's going to be dramatic!  Why don't you suggest to your sister that she determine what is really the issue, and address it, so that she doesn't have to feel all that anger?  Both she and your mother will benefit that way!