A Living Trust is a revocable trust created while a person is alive, whereas a Bypass Trust is typically an irrevocable trust created at death. A Bypass Trust can be created by a Living Trust or by a Will. (Yes, a Living Trust can create a Bypass Trust, but a Bypass Trust would never create a Living Trust.)
A Living Trust is simply an ownership arrangement where property is held in the name of a “trustee” rather than in the name of the person who really owns the property. People almost always create Living Trusts for their own benefit, with the goals of avoiding probate, addressing the possibility of future incapacity, and keeping matters private.
Normally, the person who creates a Living Trust names himself or herself as trustee and as beneficiary. Upon that person’s death, all or a portion of the property which remains in the Living Trust passes according to the terms specified in the trust agreement.
Bypass Trusts are most often created when a husband or wife dies in order to save taxes when the other spouse passes away. When a married person dies and leaves everything to his or her spouse, that surviving spouse may then be too wealthy to pass everything to their beneficiaries tax free. Being “too wealthy” typically means the married couple is worth over $5,430,000 (as of 2015). The Bypass Trust is a way to shelter the first spouse’s $5,430,000 exemption from taxation when the surviving spouse dies, thereby doubling the amount that can be left tax-free to $10,860,000.
Bypass Trusts do have non-tax benefits though, and for some people, saving taxes is not the motivating factor in creating one. For instance, Bypass Trusts protect the trust property from creditors’ claims, and they allow the deceased spouse to direct where the trust property passes when the other spouse dies.
There are some exceptions to the statements contained in this answer. For instance, Bypass Trusts are not always created at death. Some wealthy people create them during life, and other people use their estate tax exemptions for different purposes rather than the creation of a Bypass Trust. Also, in answering your question, I have assumed that when you said “Living Trust,” you meant the standard type of revocable trust people across the country regularly create and not another unusual type of trust which may be created while someone is living.