Every trust has beneficiaries. A beneficiary is a person whose benefit the trust was created.
There are generally two kinds of beneficiaries - income beneficiaries and principal beneficiaries. Income beneficiaries, in general, are entitled to receive the income that the trust assets produce. For example, if the trust owns CDs, then the income beneficiaries are entitled to the interest. If the trust owns stock, the income beneficiaries are entitled to the dividends. If the trust owns rental real estate, then the income beneficiaries are entitled to the rental income.
In your typical "avoid probate" revocable living trust, the Settlors (Mom and Dad, for example) are also the income beneficiaries of the trust. Since the revocable living trust is, for tax purposes, a Grantor Trust, then the trust does not need to file a return or pay tax. The Settlors report all trust income on their personal return, just as if they owned the property in their name.
Each trust also has principal beneficiaries. In your typical revocable living trust, the Settlors designate the principal beneficiaries. Perhaps this would be the children of the Settlors. It's worth noting that most revocable living trusts allow income beneficiaries to use principal for themselves while they are alive.
One of the benefits of using the revocable living trust is that, when the Settlors die, the Successor Trust may immediately disburse trust assets to the principal beneficiaries outside of any court or government-supervised probate (or in Louisiana referred to as "Succession") proceeding, effectively eliminating costs and delays that other families incur when they are required to settle the estate through the court system.