Do my mother's belongings become the children after she dies?

If your mother died unmarried with no will, then all of her assets (other than assets titled jointly with another, with a payable on death designation, or with a beneficiary designation) should be distributed to her children, in equal shares. If she has any deceased children, then such deceased child's share would pass to his or her descendants. Depending on the total value of her estate, and depending on whether or not her boyfriend will cooperate, a probate proceeding may be necessary. Note that, if your mother had a will in place at the time of her death, then her assets should ultimately be distributed according to the terms of her will.