India has had thermonuclear weapons since 1998, atomic weapons since 1974, and Intercontinental Ballistic Missile capability since 2012.
The Indian nuclear program was started almost immediately after independance by nuclear physicist Homi J. Bhabha, but kept consistantly on the backburner by Prime Minister Nehru (due to his pacifist leanings) until his death in 1965, and thus India only developed its first device in 1974, during the government of Nehrus rather less pacifistic daughter Indira Gandhi, as a tech demonstration known as the "Smiling Buddha". New Delhi did not immediately create an arsenal for lack of a delivery system and also for lack of wanting to cause alarm in the Cold War environment going on at the time.
In 1983, India began their ballistic missile program, culimnating in the Agni series of ballistic missiles. Agni I is an SRBM (short-range ballistic missile), while Agni II, III, and IV, are IRBMs (intermediate range ballistic missiles. The Agni V is India's first ICBM, though they are also developing the Agni VI and the Surya, which can go even further.
In 1998 however, India while under the rule of a more nationalist right leaning party, announced their unambiguous presence in the nuclear club by detonating five nuclear devices in the same desert test range used for Smiling Buddha. These devices included one low yield tactical (battlefield) weapon and one thermonuclear device. This was seen as a major intelligence coup on the part of India, as no one, not even the US intelligence community with its network of spy satellites could detect the weapons being built and moved to the test site. However, tit for tat nuclear tests by Pakistan signaled their entry into the nuclear club too and was one of the instigating factors for the Kargil War.
Also developed, with the Russians, is the nuclear-capable Brahmos cruise missile, currently the fastest cruise missile in the world.
Currently, India is developing submarine-launched ballistic missiles in order to complete its nuclear triad.
The name is in reference to the third eye of the Hindu god Shiva, which is said to lay utter destruction to all in its path should it ever open.