A trust is a separate entity in which the legal title to property is vested in a trustee who manages it for the trust and the beneficial title is held by the beneficiaries. The beneficiaries are the ones who ultimately get the distributions from the trust. Typically there is a trust document that specifies how the trust will work and when distributions will be made. The idea of a trust is that the person who sets it up wants to give away some or all of his property to others (often the spouse or kids, but it could be anyone) but only when certain conditions have been met. So he sets up a trust and gives instructions to the trustee on when the trustee should give that property to the other folks--the beneficiaries.
A conservatorship occurs when someone is incompetent to handle his/her own financial affairs. That could be becaus the person is a minor (and the law considers minors to be incompetent to handle their own finances) or because the person is an adult but mentally unable to make financial decisions. A court will then appoint a conservator to take over the management of the incompetent person's assets for the benefit of the incompetent person. The conservatorship ends when the incompetent person dies or the incompetency ends (i.e. the minor becomes an adult or the person regains the mental ability to handle financial stuff).
When someone dies, if they leave assets to minors, it's a good idea to do it in a trust so that you can specify how the trust will work and when the kid gets the assets. If no trust is set up, then a conservator is appointed (which may just be the kid's parent) to manage the assets the kid inherits untl the kid becomes an adult.