This is a very valid question, but not one that can be answered in a few words or even in a few paragraphs. It requires explaining and demonstrating multiple background concepts before one can begin to answer something this complex.
However, to summarize for the benefit of Stack Exchange users, assuming you already understand how ragas and rhythm function in Indian classical music, there are two major types of improvisation. Free improvisation (without the tabla or a composition) and improvisation around a composition with the tabla.
Free improvisation (alap) is further divided into vistar (exploring the melody of a raga at a gentle pace, showing its various notes in different facets), jod (catchy, lilting patterning of notes as the pace of the performance increases), and taan (mathematical sequences of notes that are performed at double-speed) to bring the alap to a conclusion. The video below demonstrates using Raag Bageshree.
Traditionally the alap section of a performance forms a large part of it and can go on for up to 40 minutes to an hour or more. It is then followed by improvisation around a composition, called bandish or gat. Here the rules of rhythm must be followed when improvising. A composition is set to a specific rhythm cycle, meaning that each syllable of the composition falls on a particular beat of a particular rhythm cycle. When improvising around the composition, the artist must make sure to come back periodically to the composition at the correct beat. Here is a small demonstration of how that is done around the composition eri aali piya bina in Raag Yaman.
Improvisation around a composition
Again, improvisation around a composition can come at different speeds; and it can use different vocalization techniques in the case of vocal music.
This explanation focuses on Hindustani classical music, but the essential features are somewhat similar in both Carnatic and Hindustani classical music.
Source: The explanations and the links are from my website on Indian classical music, which gives a more comprehensive explanation. The website is not a commercial operation. It is a free educational service with no ads. The videos, like the website, are my own (they are demo videos created for use on the website).
Disclaimer: My singing is good enough to provide a demo, but please note that I am not a professional singer.