(It is recommended that you read The brilliance of Panini before this article)

रामः पठति – Rama studies. If it happened in the past, you would say, रामः अपठत् and if it is going to happen in the future, it would be रामः पठिष्यति. You can see the transformations of the word पठ and you would think that is the root word. According to Panini, the root is पठं. That is the dhatu or the verbal root. That root has some extra attachments which is then transformed through a set of operations, which we can call a Paninian Prakriya, to the commonly used form. We don’t use all the letters in a dhatu while speaking or common usage and so the dhatu has to be cleaned up.
Given that there are around 1943 dhatus, how do you come up with rules that can be applied  to derive the common form? That’s where you see the brilliance of Panini, who built on top of the work of other grammarians. Paninian grammar is the only complete, explicit, rule-bound grammar of any human language and studying this grammar gives insight into the mind of our rishis.
To give concrete examples of how the Paninian prakriya works, let’s look at a set of rules called it-saṁjñā. As we saw, the dhatu has a bunch of extra additions and they need to be cleaned up. In it-saṁjñā, Panini specifies a sequence of rules which are applied to the dhatu. Each rule attaches a flag to certain letters in the dhatu. Once you run through all the rules, the last rule says, drop all the letters that have been marked. With this, the dhatu is cleaned up.

Few rules of it-saṁjñā

Here we will look at few of the rules and use simple examples. Once these are laid out, we will use some complex examples and see how they all connect as an algorithm. If you don’t get these concepts, it’s fine. How all these play together will be explained in the final section.
Rule 1: उपदेशेऽजनुनासिक इत् It means that any vowel that is nasalized, gets it-saṁjñā For example, take the dhatu धूपँ and split the words as ध् ऊ प् अँ. In this list there is a nasalized vowel, and it is अँ, so the अँ gets the it-saṁjñā. Think of it as underlining the अँ with a red color marker for now.
Rule 2: हलन्त्यम् This means that if there is a consonant (hal) that is at the end (antyam) then it gets the it-saṁjñā. To take an example, इङ् can be split into इ ङ् and since ङ् is a consonant, it gets the it-saṁjñā. What remains is इ. Also take a look at the प्रत्याहारः rules  to see what hal means.
Rule 3: न विभक्तौ तुस्माः means that there is an exception to rule #2. This rule falls into a category known as उत्सर्ग-अपवादः  The rule says, yes, you can drop the consonant as per rule #2, except when the consonants are त्, थ्, द्, ध्, न्, स्, or म् (the expansion of तुस्माः) or is the end of a विभक्ति . Take the word तस् = त् अ स्. Due to rule #2, स् should be dropped. But स् is at the end of विभक्ति प्रत्यय and hence is not marked as it-saṁjñā.
Rule 4: आदिर्ञिटुदवः ञि, टु, and डु (ञिटुडवः) are marked as इत् if it is at the beginning (आदिः) Take the dhatu डुदाञ् = ड् उ द् आ ञ्.  Here डु receives the इत्-संज्ञा by this rule.
Rule 5: षः प्रत्ययस्य The letter ष् (षः) found at the beginning (आदिः) of a प्रत्यय  is given the marker इत्. For example, in षाकन् = ष् आ क् अ न् , the first letter ष् receives the इत्-संज्ञा.
Rule 6: चुटू The letters च्, छ्, ज्, झ्, ञ्, ट्, ठ्, ड्, ढ्, or ण् (चुटू) found at the beginning (आदिः) of a प्रत्यय are marked इत्. For example, in णिच् = ण् इ च्. Here ण् receives the इत्-संज्ञा because of this rule.
Rule 7: लशक्वतद्धिते The letters ल्, श्, क्, ख्, ग्, घ्, or ङ् found at the beginning of a प्रत्यय, but not at the beginning of a तद्धित प्रत्यय are marked इत्
Running the above the rules underlines various letters for इत्. The last rule is like the garbage collector in programming languages. It finds all those marked letters and drops them.
Rule 8: तस्य लोपः — All those elements that received the marker इत् (तस्य) disappear (लोपः)

Examples with explanations

Now that we know all the rules, it would help to take some examples and run them through this sequence and see the transformation of a dhatu.

1. णमुँल् – Breaking down into individual letters, you get ण् + अ + म् + अँ + ल्  Now apply rule #1 and since there is a nasalized vowel (अँ), mark it with red color. Then go down the list of rules and see which one matches.  The next one that matches is #2 and mark ल्  with a red color marker. Scan through the rule set and we find that #6 applies as well. So we mark ण्  with our marker. No other rule applies here and so we come to rule #8 which says, drop all those letters which were marked. Doing that gives us अम्
2. ञिष्विदाँ – Breaking down that again, you get ञ् + इ + ष् + व् + इ + द् + आँ . Going through the rule set, rule #1 applies you we mark आँ with a marker. Then going down, we find that rule #4 also applies, marking ञ. Finally, we come to rule #8 and drop those two letters leaving us with ष्विद्
3. Let’s do one more. ध्वम्. This breaks down into ध् + अ + व् + म्. Rule #2 applies here and we mark म्. Then the third rule kicks in and says, hold on. You cannot mark it and so we are left with  ध्वम् itself.

The grammarians of India looked at Sanskrit used in scripture and for speaking and then came up with the grammar which was used to build up the rules. The it-saṁjñā is just one of the many such rules.
There is one question though. If you are using the word पठ, then why bother writing it as पठं. and then apply all these rules to drop few characters. That is a topic for another post.

PS:

1. Based on the lectures by Varun Khanna at Chinmaya International Foundation and my Sanskrit teachers.
2. Taking धूपँ and splitting up it into ध् ऊ प् अँ is called anupurvi
3. The dhatu णमुँल् is called sa-anubandha dhatu and the cleaned up version अम् is called the niranubandha dhatu