If my mother passed with no Will, how do I go about transferring her property into my name?

Each state has procedures to transfer title to property after a person dies without a will.  Some procedures are more complicated than others.  For example, Florida has a reasonably complex series of laws covering a person's homestead property and requires opening a full administration estate in the Probate Court to transfer that title.  On the other hand, Florida has fairly simple and quick procedures for transfering title to an automobile. 

The first question you need to answer is "who is entitled to the property?"  This depends on the law of descent and distribution in your state.  If you are the only child and your mother had no spouse at the time of her death, it is reasonably safe to predict that you are the person entitled to the property no matter what state you live in.  On the other hand, if there are other children or a spouse, this gets more complicated.

I suggest that you make a list of your mother's property and then call the Probate Court in the county where your mother lived.  While the courts will not give legal advice or represent you, the clerks or magistrates will often tell you where to get information specific to your circumstances.  That court's web site may also have useful information, and your state's bar association web site often has useful information.