The answer depends on the law of the state where your sister died. Most states have "rules of intestacy" that say who gets the assets of a person who died without a will. Most states follow the same general system of the closest "next of kin."
In most states the progression is to a spouse first, if no spouse then to children, if no children, then to parents, if no parents then to siblings, if no siblings then to nieces and nephews. If the estate is to go to a group of people (children) and one or more of that group has died, and leaves children, then that person's share usually goes to their children.
In your situation, it sounds like your sister has children. Most likely they would be her heirs. If she has a child who has died, then that deceased child's share should go to that deceased child's children. You state that you are the deceased person's sister. You are probably not an heir and will not get anything, if your sister has living children or grandchildren.
Someone has to volunteer to administer your sister's estate. That person should seek the assistance of a local attorney to ensure it is done right.
I am so sorry for your loss. I do not practice in GA so I will give you a general answer.
That being said, I will answer your question with a list of questions. Sorry about that, but it must be done. The first thing you need to find out is if your mother has a will or a trust. A valid Will or Trust will tell you exactly where the money is supposed to go. If the will or trust includes your sister's children, then they will be included if the assets pass to the heirs through the will or trust.
If the assets are in your mother's name and there is no will or trust, the assets pass by the law of default. This law is also called the law of intestacy. Generally, the assets pass to the children and to the grandchildren of any deceased child. If the law of intestacy controls, the assets will pass to the family through the probate process. It just passes to the family without a will.
You must then check the ownership on your mother's assets. If the assets are in her name alone, the assets will probably need to go through probate. If the assets are in the name of your mother's trust, the assets will pass to the beneficiaries spelled out in the trust. Finally, and most importantly, if your mother owned her property jointly with someone else, the assets pass to the remaining owners. The assets never get to the children or grandchildren because the assets go directly to the other owners.
I strongly urge you to take the information you have and find the answers to my questions. Then go to a local attorney and the attorney will be able to tell you the beneficiaries to your mother's estate.
However, the law which determines where assets go after someone dies, the law of intestacy, decides where the money goes.
Ilene's response was very throrough and Georgia law follows what she said there.
Your interpretation of the information is also correct. One-third to each each living child and one-third to be divided among the child of a deceased child. If your mother wants anything different to happen, she must do a will or trust. If the house is her only asset, she could also transfer a remainder interest now, while she is alive, and retain a life estate for herself, thereby avoiding probate for the house.
Georgia intestacy law says that if an unmarried person dies without a will, her assets will be distributed in equal shares among her children. If one of those children died first, that child's share will be divided among his/her children.
If any of the individuals who inherit are minors (less than 18) or are not competent at the time they inherit then a Conservator may need to be appointed for that individual.
Having a well drafted Will can avoid the need for a Conservator, and allows for the individual to provide for her family members however she wishes rather than being bound by intestacy law. You may find the article I have written about Wills, at http://www.elderlaw-lacey.com/articles/legal_foundations.pdf, to be helpful.
Your question did not mention where your mother lives and whether or not she is married. Further, you did not mention whether or not your mother has a last will and testament or a trust. Many questions in these posts appear simple but they often raise questions because of the limited facts. For instance, you don't state what property your mother has and how this property is titled. If property is held jointly with rights of survivorship then the surviving joint tenant receives the property upon the death of the other tenant. If property is held not as with rights of survivorship but as cotenants in the death of the cotenant will not be distributed to the surviving cotenant in the absence of a last will and testament or a trust directing that distribution. In addition, you do not state whether or not your mother has a last will and testament or a trust. For purposes of answering your question I and one to assume that your mother lives in Georgia because your question mentions Tallapoosa GA. Further, since you did not mention your mother's husband I am going to assume that she is not married for purposes of this answer. If your mother has a last will and testament or trust then the terms of those documents will control the distribution of your mother's property. If your mother does have a last will and testament or trust then you need to consult an experienced estate planning attorney to determine whether or not your deceased sister's children would receive her share upon the death of your mother. As a general rule and under most wills I have prepared or seen the children of your deceased sister would receive their deceased mother’s share of property under the last will and testament of your mother. This, however, may not be the case and only an experienced estate planning attorney can tell you this. If your mother dies and does not have a last will and testament that is referred to as dying intestate. In the case of someone who dies intestate (i.e. without a will) then the Georgia rules of inheritance apply. Under the Georgia rules the children of your deceased sister will stand in your deceased sisters "shoes" as it relates to any inheritance from your mother. In other words, your deceased sister's children will receive their mother share of your mother’s (i.e. their grandmother’s) estate. Any share that the children of your deceased sister receive will be divided and approximately equal shares among those children.