How can you use roots of in Arabic to form different words?

Arabic is a very formulaic language, and you can use this to your advantage! Arabic words are derived from basic root words, usually comprising of three letters. Let's use the example of كتب ., which is a very common root. كتب on its own is a past tense verb, meaning "he wrote." Like other languages, you can also alter it slightly to change the person and the tense of the verb. For example, اكتب means "I write".
What is special about Arabic is that its flexibility allows you to form nouns from these roots too. Say if you add an "ا" after the first letter of the root, your newly created word is the "doer" of the original verb. Using the example above, كاتب would therefore translate an author or a writer. Furthermore, you could add a "م" before the root to indicate a place which is related to original meaning. So, مكتب is an office, and its feminine equivalent مكتبة is a library. There are many other such examples, but this hopefully gives you a taster of how you can form several different words from one singular root.