Fundamentally, a trust exists where one person holds property on behalf of another. So it is a type of fictitious enttiy used to hold and own property. The person who sets it up is called a "settlor," the person who legally owns the property is called the trustee, and the beneficiary who equitably owns the property is called the "beneficiary."
A trust may be declared by a person during his or her lifetime, or as part of a valid last will and testamenty. A trust set up during the Settlor's lifetime is called an "intervivos trust," or, a "living trust." A trust set up in a will is called a "testamentary trust."
A trust that has terms and conditions which can change, is called a "revocable" trust. One that has terms which may not change is called "irrevocable." Necessarily, all testamentary trusts are irrevocable, because the settlor has already died. Likewise, in general revocable trusts become irrevocable when the settlor dies. On the other hand, intervivos trusts may be revocable or they may be irrevocable.
Generally "revocable trust" and "revocable living trust" and "living trust" are all terms for the thing.
Oftentimes revocable trusts are used as will supplements because successor trusteeship may be immediately assumed under the conditions stated in the trust, whether for death or incapacity, and so offers a form of continuity of adminstrative leadership that the will probating and fiduciary appointment process does not. Also lawyer fees for handling trust property will oftentimes be less than fees for handling estate property adminstered in court. Hoewever, revocable trusts are advisedly used in tandem with and as supplements to wills, and not as "replacements" as sometimes said.
Persons who have modest property ownership may not benefit from use of trust planning. Likewise, persons whose estates are at the size or complexity which may benefit from use of trust planning, will tend ot use revocable trusts rather than irrevocable trusts. Irrevocable trusts are mostly related to high net worth individuals with interests that may require comparatively more expensive planning for purposes of tax planning and asset protection.
A "living will" is a misnomer, a term often confused with these, hnowever, it is netiher a trust nor a last will and testament. It is a form of health care directive, another subject entirely.