Tens of compositions are played on the tabla and creating an exhaustive list would be difficult. Not only because of the sheer number of them, but also due to the blurry boundaries of some of them. There is a considerable amount of overlapping depending on who you ask. However, this should not be a hurdle in our understanding.
Kaida / कायदा
Kaida comes from the word ‘Kaid’, which means to cage or bind. Therefore, a kaida is one which is played under certain boundations and rules. These are played only on those taals in which solo playing is popular. The whole structure of the taal, including its taali, khaali etc. has to be taken care of while constructing a kaida. Kaida is made up of those bols which can be mixed and matched to form various Palte. Palte (पल्टे) are the various derivatives of a kaida which are formed by using the bols of the kaida in a different manner. A kaida ends in a Tihai. Here are some examples of kaidas in teentaal.
Peshkara / पेशकार
Peshkara comes from the word ‘Pesh’, which means ‘to show’. This is the first composition that is played in a solo performance and shares many characteristics with the Kaida. Therefore, it is essentially an elegantly crafted kaida with its own palte. Also, a Peshkara is played in the Madhya Laya. Some bols like ‘DhaKra’ and ‘SKaDha’ are especially common in a Peshkara.
Tukra / टुकड़ा
Literally translating to ‘a piece’, it is a very common compositions that is played on Tabla. While technically every composition is a ‘piece’, a Tukra has some specific qualities:
A tukra is usually not as long as a Paran, and usually does not span more than one or two avartans. Usually finishing with a Tihai, it is played between longer compositions like Kaidas.
Paran / परन
While originally it is a thing related to the Pakhawaj, it is widely played on the tabla too. It can be described as a special, impactful, and long Tukra. One of the defining qualities of a Paran is that the bols go about repeating.
A paran usually spans four or more avartans and ends in a Tihai, just like a Tukra.
Uthan / उठान
Uthan is a kind of hybrid of tukra and Paran. It is more impactful than a tukra but smaller than a Paran. It is usually played at the very beginning of a dance or solo session (as is evident from the root word – utha – meaning ‘to pick up’). This is usually played in the Madhya Laya but this isn’t compulsory.