Texas, like other jurisdictions, treats debt as the responsibility of the individual who incurred it. As such, children don't necessarily inherit debt after a parent's death. However, a parent's debt does need to be paid out of the parent's estate. This effects children because it reduces the amount of inheritance children receive. That said, however, children do not typically become personally liable.
Debt is typically personal to the individual who incurred the debt, particularly credit card and student loan debt. In some community property states, marital debt can be inherited by a surviving spouse. Texas is a community property state. It's important to note, however, that Texas is not a community debt state, meaning one spouse does not inherit the other spouse's debt after death simply because of Texas' community property state status.
When a person dies, either with or without a will, a Texas probate court assists in settling the decedent's financial affairs. The responsibility of the person appointed as executor, or personal representative, is to pay the decedent's debt using funds from the decedent's estate. This is done by sending notice to the decedent's creditors, then paying the creditor's claims. After the decedent's debts are paid, whatever remains goes to his heirs, either according to his will provisions, or according to Texas laws of intestate succession.
When a decedant dies with debt, his debt must be paid out of estate assets, thus reducing the amount his heirs inherit. As such, it's not true that children inherit debt. They are, however, effected by a parent's debt because, if a parent's estate assets are totally depleted due to debt, the children receive no inheritance. Children should know, however, that if their parent's estate is insolvent, meaning there's not enough money to pay the parent's creditors, the children are not responsible for paying the debt with their own money.
One rare exception regarding debt liability concerns an executor's responsibilities. Typically, if an executor fails to notify creditors of probate, he may become personally liable. Thus, if a child accepts the role of executor of his parent's estate, it's possible he might be liable for any debt he failed to pay out of the estate's assets.