There are two major Indian classical music traditions, Hindustani (North Indian) and Carnatic (South Indian). The link is focused on the former. I'd like to add some (general, not just rhythmic) notes on the latter.

1. Both derive from the Sāmaveda, and took their modern forms beginning from the 15th century CE. What would eventually become Hindustani music took on Persian and Arabic influences, whereas Carnatic music—as the name suggests—is mostly limited to South India and therefore relatively insulated from the cultural changes in the North, and is considered closer to ancient Indian music.

2. Perhaps the most important difference between Indian and Western classical music is that the former is relative, not absolute. The main artiste picks a comfortable śruti, or tonic: males typically choose B to D, solo instrumentalists typically tune their instruments to D - F, and female artistes choose F# to A (and an octave higher). This is just a rule of thumb; some instrumentalists (flautists, for instance) prefer C. Note that I refer to only the note; the precise frequency/octave is irrelevant insofar as defining śruti is concerned. Once the main artiste has selected their śruti, all other accompanying instruments are tuned to the same. This is why ICM notation is solfège-based, staff notation does not work for ICM, and why Indian classical musicians have a taṃbūrā (could be electronic, too) on stage.

3. The ICM solfège is called swara. There are seven of these, called sa, ri, ga, ma, pa, dha, ni, just like in the West (do-re-mi-fa-so-la-ti). Similarly, they expand to twelve when accounting for semitones.

4. The basic melodic framework for ICM is the rāga; loosely put, this defines a certain scale that a composition, or parts of a composition, or some improvisation is set to. rāgas also define the gamakas, or ornaments for their scales; this is fundamental to ICM. There exist notationally equal scales with and without gamaka, but they are considered completely different rāgas.

5. There are loose analogues between ICM rāgas and Western modes. For instance, the Ionian mode is equivalent to Śankarābharaṇaṃ in Carnatic music, or Bilāval in Hindustani music.

6. Carnatic music rāga classification has an interesting tree structure. There are 72 'roots', called mēḷakartā, which have a complete scale of 7 notes ascending, and 7 notes descending. Delete some notes, or reorder them, or introduce one or more notes from another scale, and you get a veritable sea of janya rāgas.

7. Carnatic music has a few classifications for its rhythmic patterns. The suladi sapta tāla system is most commonly taught to beginners. In this, there are three angas (or parts): laghu, dhrutam, and anudhrutam. laghus can have 3, 4, 5, 7, or 9 beats (or akṣara), called jātis. There are seven arrangements (ergo, sapta) of angas under this system, and with five jātis, there emerge 35 talas with different akṣara sums for each. The typical 8-akṣara rhythm (again, similar to Western music) is called ādi tāla, or catusra jāti triputa talā.

Some exemplary Carnatic music:

Vocal: Ramakrishnan Murthy, ānanda naṭana prakāśam, Muttusvāmi Dīkṣitar, Kēdāram, Misra Cāpu tāla: https://www.youtube.com/watch?v=Eu-tFbGpoTQ

Violin solo: Charumathi Raghuraman: https://www.youtube.com/watch?v=lESyDHH_E7Q

Vīṇā solo: Ramana Balachandran: https://www.youtube.com/watch?v=SgW3j9OVpJo

Flute solo: J.B. Sruthi Sagar: https://www.youtube.com/watch?v=v6InFwzX8kk