Investiture, from the Latin (preposition in and verb vestire, 'dress' from vestis 'robe') is a rather general term for the formal installation of an incumbent (heir, elect of nominee) in public office, especially by taking possession of its insignia. The term is normally reserved for formal offices of state, aristocracy and church.
In the feudal system investiture was the ceremonial transfer of a fief by an overlord to a vassal. The lord invested the vassal with a fiefdom, by giving a symbol of the land or office conveyed in return for an oath of fealty. From feudal times up to the present, the term has been used in ecclesiastical law to refer to a cleric receiving the symbols of spiritual office, such as the pastoral ring, mitre and staff, signifying transfer of the office.
As the insignia can include the formal dress and adornment (robes of state, headdress etcetera) the etymology refers to, but also other regalia in the widest sense, such as a throne or other seat of office, the word is a convenient generic term, also for such more specific cases as coronation (see that article and regalia for more on such ceremonies) and enthronement, though these are also used (rather imprecisely, by analogy) in such extended sense.

Secular usage
The term is used to describe the installation of individuals in institutions that usually have been extant from feudal times. For example, the installation of heads of state and various other state functions with ceremonial roles are invested with office. Usually the investiture involves ceremonial transfer of the symbols of the particular office.
Judges in many countries, including justices of the Supreme Court of the United States, are invested with their office. American justices typically take two oaths: one to uphold the Constitution of the United States, and the other to apply the principle of Equal Protection to the rich and the poor (source: [1]). Likewise, university presidents, rectors and chancellors are invested with office.
In the United Kingdom, around 2,600 people are invested personally by The Queen or a member of the Royal Family. A list of those to be honoured is published twice a year, in either the New Year's Honours List or The Queen's Birthday Honours List. Approximately 22 Investitures are held annually in Buckingham Palace, one or two at the Palace of Holyroodhouse in Edinburgh and one in Cardiff.