crypt

In architecture, a crypt (from the Latin crypta and the Greek κρύπτη, kryptē; meaning concealed, private) is a stone chamber or vault beneath the floor of burial vault possibly containing sarcophagi, coffins or relics.
In more modern terms, a crypt is most often a stone chambered vault used to store the deceased.  Crypts are usually found in cemeteries and under public religious buildings, such as churches and cathedrals, but are also occasionally found beneath mausolea or chapels on personal estates.  Wealthy or prestigious families will often have a 'family crypt' or 'vault' in which all members of the family are interred.  Many royal families, for example, have vast crypts containing bodies of dozens of former royals.  In some localities an above ground crypt is more commonly called a mausoleum, which also refers to any elaborate building intended as a burial place, for one or any number of people.

Originally crypts were typically found below the apse of a church, such as the Abbey of Saint-Germain in Auxerre, but were later located beneath naves and transepts as well.  Occasionally churches were raised high to accommodate a crypt at the ground level, such as Saint Michael's church in Hildesheim, Germany.

Burial Vault
A burial vault is a structural underground tomb.

It is a stone or brick-lined underground space or 'burial' chamber for the interment of a dead body or bodies.  They were originally and are still often vaulted and usually have stone slab entrances.  They are often privately owned and used for specific family or other groups, but usually stand beneath a public religious building, such as a church, or in a churchyard or cemetery.  A crypt may be used as a burial vault.

Sarcophagus
A sarcophagus is a box-like funeral receptacle for a corpse, most commonly carved in stone, and displayed above ground through they may also be buried.

Sarcophagi were most often designed to remain above ground, hence were often ornately carved or elaborately constructed.  Some were built to be freestanding, as part of an elaborate sealed tomb or series of tombs, while others were intended for placement in crypts and remain accessible to view.

Relic
In religion, a relic is a part of the body of a saint or a venerated person, or else another type of ancient religious object, carefully preserved for purposes of veneration or as a touchable or tangible memorial.  Relics are an important aspect of some forms of Buddhism, Christianity, Hinduism, Shamanism, and many other religions.  The word relic comes from the Latin reliquiae, meaning "remains" or "something left behind" (the same root as relinquish).

Mausoleum
A mausoleum is an external free-standing building constructed as a monument enclosing the interment space or burial chamber of a deceased person or persons.  A monument without the interment is a cenotaph.  A mausoleum may be considered a type of tomb or the tomb may be considered to be within the mausoleum.  A Christian mausoleum sometimes include a chapel.

Sources:

http://en.wikipedia.org/wiki/Crypt
http://en.wikipedia.org/wiki/Vault_%28tomb%29
http://en.wikipedia.org/wiki/Sarcophagus
http://en.wikipedia.org/wiki/Relic
http://en.wikipedia.org/wiki/Mausoleum