A Qualified Domestic (QDOT) Trust is a very specific trust designed to address a somewhat common situation: what happens when the spouse of a non-US citizen dies while leaving significant assets to the non-citizen survivor.
Normally, thanks to the Martial Exemption to estate taxes, when two citizens are married and one dies, their assets pass automatically to the surviving spouse without estate tax being paid. The estate tax is deferred until the death of the survivor. However, this does not apply if your surviving spouse is not a citizen. Under these circumstances, your estate tax would have to be paid upon receipt of the estate.
However, a QDOT trust can get around this tax problem. Properly set up, a QDOT trust makes it so that, upon your death, your assets pass into the hands of the trust, with the surviving non-citizen receiving the benefits from the trust. Then, upon the death of your non-citizen surviving spouse, the assets pass to your relatives as normal and the estate tax is paid then, much like it would with two citizens.
One significant restriction on a QDOT trust is that the trustee – the person who controls the trust – must be a US citizen. Further, for estates over $2 million, one of the trustees must be a US bank specifically, so that the IRS has a guarantee that the taxes will be eventually paid. Any income earned from the trust is subject to income tax, but not estate tax. Also, when there's any distribution from the principal of the trust (the assets initially used to create it), estate taxes are due immediately on what was distributed. However, there are hardship exemptions to this rule.
That said, the need for a QDOT trust can be avoided if your survivor simply becomes a citizen before the taxes are due. So, while these trusts can be quite useful under very certain circumstances, in most cases the need for them can be avoided. None the less, if you are in this unique situation, a QDOT Trust is a good choice for helping your spouse avoid paying your estate taxes.