The Tabla is a set of two drums which evolved from a single double-ended drum similar to the Pakhawaj, which is still played in India today. The right hand drum bears the name Dayan or Tabla and is the higher pitched of the two drums.  It is played with rapid tapping of the fingers with the index finger left in contact with the edge of the skin to achieve higher harmonics.  The left  drum, called the Duggi or Bayan, is the deeper sounding drum which makes the characteristic deep bending (gulping) sound.  This tone is created by striking the skin of the Duggi with the fingers while pushing down on the skin with the palm of the same hand.
The Pakhawaj evolution took several other paths in different parts of the country giving India a rich array of drums differentiated by body shape and size which all share the same basic characteristics of head construction.  The Mirdangam is played in the south of India.  This drum is still very similar to the Pakhawaj of today except that the Chati, a second piece of leather which extends over the resonant skin (Maidan), is thicker and extends all the way to the syahi.  The dholak is a more basic style drum, still with two different sized heads, which often does not have syahi on either skin.  It is used to accompany male vocalists in Indian folk music rather than classical style.  The dholki is the same drum with a syahi on the smaller head.  This drum is used to accompany female folk vocalists.  A dholki which is bolt tightened with a syahi on the bass skin is called a Nal which is used in pop music. 

The Khol is a Pakhawaj relative used in temples and Hindu devotional songs.  (Popularized by the Hare Krishnas)  This drum is often ceramic (most other of the Pakhawaj family are typically wood) and has a bulge in the center.  The bols of the khol usually mirror the consonant sounds of the song they accompany.

Tabla, Mirdangam and Pakhawaj rhythms are played in cycles called "Tal" to accompany other musicians.  The audience participates in the counting of the Tals throughout the performance by making gestures with their hands to measure the occurrence of the Sam beat which is the start of the rhythm cycle.  During improvisations the musicians play separate rhythms for a period of time finally converging their respective rhythms on the same Sum beat.  It is this overlaying and then merging of different rhythms that makes the audience applaud during performances of Indian classical music.

Classical percussion styles are passed down through lineages (gharana) of teachers as a vocal tradition.  Occasionally, singers will even vocalize these spoken approximations of the Tals in their performance, sometimes "dueling" with the tabla player who replicates the rhythm in answer to the singer.

Indian Drums in the Rhythmuseum

Zakir Hussain
More on Tabla
Ghatam - TH Vinayakram
Kanjira- V Selvaganesh

Indian Percussion
The Pakhawaj has one syahi on the smaller skin and the bass skin is resonated with a dough-like concoction which the drummer applies during performance. 
The Mirdangam is a Pakhawaj related drum played in the Carnatic music of South India.
The khol (pictured above) is used in Hindu devotional songs. The thavel (pictured below) is a drum played primarily in Shiva temples of South India.  It is played with thimbles on the fingers of the right side and a stick on the left side.