Vedas are eternally existing sounds. They are Apaurusheya (no creator). They exist as the breath of Brahmam itself.
Now, any language is a combination of 4 things - sounds, grammar, meaning, and script (optional).
The language of Vedas / SanatanaDharma / Hinduism is called Sanskrit. Out of 6 Vedangas (auxilary), 4 deal with this language :
- Shiskha - sound/pronunciation
- Vyakarana - grammar/sentences
- Chanda - meter/prosody
- Nirukta - meaning/dictionary
For most languages, the sound-meaning associations and written script can change over time e.g. the word 'thou' became 'you' in English.
Except for Sanskrit, whose sound-meaning associations never change - because they are of divine origin.
While the written script changes over time - because it is of mortal origin, e.g. Brahmi became Devanagari.
The story behind how Sanskrit grammar (Vyakarana) was formulaized (literally made into forumlas or Sutras) by Panini is an interesting one.
Panini and Vyagrapadha rishis did tapas towards Shiva, to watch his divine Tandav. When he appeared before them, he also played his dumroo. This created 14 primordial sounds, known as Maaheshwara Sutras. From this, Panini was able to derive formulas, and compiled into a magnum opus known as Ashtadhyayi. These were extremely terse. So Patanjali explained them in detail in Mahabhashya.
Fun fact: Vyagrapadha (tiger's feet), is the reason for the name of a town in TamilNadu called Sirupuliyur (puli = tiger). Source : Krishna Premi Upanyasam.
There is a common misconception, usually propounded by Western Indologists, that Sanskrit 'manually evolved over time into a perfect form' and that's why its called 'Samyak Kritam' (well-done), hence 'Samskrutam'.
They even come up with nonsense theories like PIE (Proto-Indo-European language) as the forefather of Sanskrit, despite there being no civilization, country, culture, history or script associated with PIE.
Whereas Sanskrit has all of them - a civilization, country, culture, history and script.
Even despite the overwhelming number of Sanskrit origin words used in Latin and thus English (e.g. the word 'mother' morphed over time from Sanskrit word 'matru', father from pitru, brother from bhratru, daughter from duhitri.. the list is endless), despite Baltic languages like Lithuaninan sharing a ton of similarties with Sanskrit, despite the Gods of most ancient civilizations bearing resemblance to Indra, Agni, Varuna etc., their ego is unable to digest the antiquity of Sanskrit, and they end up concocting more fantasy like Aryan Invasion Theory.
TL;DR : Sanskrit is the name of a language which is a collection of sound-meaning combinations of the eternal Vedas. It's grammar was expressed in forumlas by Panini based on Shiva's 14 primordial sounds emanating from his dumroo.