A cathedral is "the Bishop's church" -- the chief church of a diocese.
The word is derived from the Greek kathedra through the Latin cathedra,
meaning throne or seat, which is the symbol of authority.
A basilica is a church with important canonical status and privileges,
usually due to its history. There are two types of basilicas:
Major basilicas have a special "holy door" and a papal throne and an
altar at which no one may say Mass except by the Pope's permission.
They are also called "patriarchial basilicas" and are associated with
the great patriarchates of the world, for ex.:
- St. John Lateran
is the cathedral of the Pope and therefore associated with him as
Bishop of Rome and as Patriarch of the West
- St. Peter's
Basilica, the Patriarch of Constantinople
Paul-without-the-Walls Basilica, the Patriarch of Alexandria
- St. Mary Major,
the Patriarch of Antioch
(the above 4 Basilicas are the original and primary Basilicas)
Lawrence-outside-the-Walls, the Patriarch of Jerusalem
- The church of
St. Francis at Assisi
- The church of
Minor Basilicas are specially linked to the papacy and are to be places
of pastoral zeal and dedication to the liturgy, including the Divine
Office ("Liturgy of the Hours"). They have a sanctuary fixture known as
a conopaeum (also called a papilio, pavilion, umbellina, or
sinicchio), a tall, umbrella-like canopy made
of alternating bands of red and yellow silk -- the colors of the papal
government -- and topped with a copper Cross, the original function of
which was to shelter the patriarch. Basilicas are also allowed to have
their own coat of arms and can have members carry a special tower with
a small bell in processions. Some famous Minor Basilicas are Ste. Anne
de Beaupré in Québec; Sacré Coeur in Montmartre, Paris; National Shrine
of the Immaculate Conception in Washington, D.C., etc.
An oratory is a place other than a parish church where it is
canonically permissible to celebrate the Liturgy. Some are totally
public, some are semi-public (such as in monasteries, hospitals,
prisons, etc.), and some are private (in private homes, private
oratories can only be set up by permission of the Pope).
Anyone can set up a chapel -- and there are many varieties of chapels,
including chapels inside larger churches. But to be able to celebrate
the Mass in a private chapel, permission of the Bishop is required. The
word "chapel" (and the word "chaplain") derives from the Latin "cappa,"
meaning "cloak," in reference to the chapel built to house a certain relic -- the half of the cloak that St. Martin
of Tour kept after giving the other half to a beggar.