Jawaharlal Nehru
 August 15, 1947 - May 27, 1964
(Indian National Congress (INC), which led the struggle for independence)

Lal Bahadur Shastri
 June 9, 1964 - January 11, 1966
(INC.  Succeeds Nehru after the latter dies of a heart attack)
 
Indira Gandhi
 January 24, 1966 - March 24, 1977
( INC.  Installed as PM by the "Syndicate" after Shastri's death, in the hope that she would be a malleable leader, and the Syndicate the "power behind the throne)

Morarji Desai
 March 24, 1977 - July 28, 1979
(As leader of the JANATA PARTY, a coalition formed to oppose Indira Gandhi during her imposition of the STATE OF EMERGENCY)
 
Charan Singh
 July 28, 1979 - January 14, 1980
(Differences within the JANATA party lead to the resignation of MORARJI)
 
Indira Gandhi
 January 14, 1980 - October 31, 1984
(Wins elections in 1979-80 after factional conflicts make the Janata Party unable to stay together)
 
Rajiv Gandhi
 October 31, 1984 - December 1, 1989
(Comes to power after the assassination of his mother in 1984)

Vishwanath Pratap (VP) Singh
 Dec. 2, 1989 - November 10, 1990
(Wins General Elections in which CORRUPTION, particularly the BOFORS deal, is a major issue.  Heads a coalition government led by his party the JANATA DAL)

Chandra Shekhar
 November 10, 1990 - June 21, 1991
(Inability of the Janata Dal-led coalition to stay together makes him PM.  He had temporary support from the INC led by Rajiv Gandhi during his short tenure.  Rajiv withdraws support in 1991, and fresh elections are called.)
 
P.V. Narasimha Rao
 June 21, 1991 - May 16, 1996
(Rajiv Gandhi's assassination during the election campaign results in a victory for the INC, and RAO now becomes leader of the party and the PM.  Completes his term in office, but loses popularity.  The Hindu nationalists, the BHARTIYA JANATA PARTY (BJP) gain support, especially after they make the issue of the mosque in AYODHYA a major campaign issue.)

Atal Behari Vajpayee
 May 16, 1996 - 1 June 1996
(Elections in 1996 result in a HUNG Parliament, no party or group of parties has an absolute majority.  The BJP is the single largest party, and its leader VAJPAYEE is invited to form a government, but cannot muster support in Parliament, is compelled to resign.)
 
H. D. Deve Gowda
 1 June 1996 - 12 April 1997
(A post-election coalition of parties, known as the UNITED FRONT (UF), comes together to try and cobble a majority in Parliament.  The UF is composed of parties that want to maintain a distance from the BJP and the INC, and many strong regional parties were an important component.  However, the UF government still depends on "outside" support from the INC)
Inder Kumar Gujral
 21 April 1997 - 19 Mar 1998
(The INC objects to some of GOWDA's policies, and threatens to withdraw support if he is not removed.  GUJRAL, a former INC member and member of Mrs. Gandhi's Cabinet, takes over.)

Atal Behari Vajpayee
 19 Mar 1998 - present
(INC withdraws support to GUJRAL as well, and another General Election called in 1998, which is won by another coalition, this time led by the BJP, also with the support of important regional parties.  VAJPAYEE becomes PM.)

Note:  Gulzarilal Nanda was caretaker Prime Minister twice in the 1960s, between Nehru and Shastri, and Shastri and Indira.