An investiture is the official appointment of an incumbent to a formal office. 

Investiture means being given a special vestment/robe/dress as an official symbol of new responsibilities. 

There are three social realms in which investitures occur: 

1.regarding the aristocracy 

2.ecclesiastical 

3.state 
Ecclesiastical (church) investitures typically refer to the ceremony at which a cleric is given symbols of spiritual office. For example, at the investiture service of a Bishop he may be given his robes and mitre etc. 

Within the church, some feel strongly that all clerics should be appointed only by the church itself. However, in many cases, other bodies, that is, lay (non-ecclesiastical) bodies have taken it upon themselves to appoint church ministers and officials. These are called 'lay investitures', but there is much controversy about the validity and necessity of lay investitures. 
Lay investiture was the appointment of bishops, abbots, and other church officials by feudal lords and vassals. No one questioned a king or noble's right to grant a bishop or abbot a fief and have him become a vassal, but the church did object to kings and nobles naming bishops or abbots. 

There was a close association between church and state which, in a sense, had started with the pope's crowning of Charlemagne in 800AD. With whom the final authority rested was unclear. In theory, the clergy, representing heavenly power, was higher than the nobility, who represented earthly power. 

Many kings were dependent upon the clergy for communication and political advice, therefore the persons acting in this capacity occupied an important position in the state. At the same time, the clergy naturally saw their roles as defending the powers of the church at the expense of the powers of the king. 

Lay investiture became the focal point for the struggle between church authority and secular authority. Did a king have the final authority in his state? Should not a monarch have the right to choose a man he trusted to be his adviser and spiritual head of his kingdom? 

The two most well-known cases that involved the issue of lay investiture were those of Thomas Becket with Henry II of England and Pope Innocent III with Henry IV of the Holy Roman Empire. 
  
  
As a teaching tool, the issue of lay investiture can be used to illustrate the church-state power struggle. This struggle eventually led to several other situations such as the rise of nationalism in the nation-state and the Reformation. In both of those cases, the kings eventually proved supreme. Royal authority and its official protection of religious dissenters was the reason that the Reformation leaders were able to finally triumph. 
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 etc) 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.

[edit] 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 apply justice equally.[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.

[edit] Ecclesiastical usage
Lay investiture was the appointment of bishops, abbots and other church officials by feudal lords and vassals. The secular ruler usually invested the elect/appointee with the insignia of his ecclesiastic office, while the Pope crowned the Holy Roman Emperor (elected by the German Electoral Princes).

The question who should invest (or more to the point, appoint) whom was the subject of an epic conflict between the Catholic Church (mainly papacy) and state (mainly the Holy Roman Empire) in the Middle Ages during the so-called Investiture Controversy.