This is a simple idea to understand:
Testamentary trusts are created in a will, and don't come into being until you die and the will is probated. Think last will and "testament" to help remember this one. Common testamentary trusts are "contingent" trusts for minors and incapacitaed individuals, bypass trusts for tax planning, and supplement needs trusts for special needs planning.
Living trusts, also called "intervivos" trusts, are created during your life, and come into being upon creation and funding of the trust. Please note that it's not enough to only sign trust paperwork; you need to actually transfer property into the trust, even if it's just $1.00, for the trust to be formed. Common living trusts include revocable living trusts, irrevocable life insurance trusts, Crummey trusts, and other irrevocable gift trusts.