Dots (and diacritics generally) have been used in writing systems for several reasons:
To mark vowels
In abjads and some abugidas, consonants are written as large characters, and vowels are written as diacritics around the main characters, frequently a dot or dots, but also dashes and other small marks.
One option for a con-writing system which I haven't seen before would be to swap this, and have the vowels be the main characters, and the consonants be indicated by diacritics.
Diacritics are used in some writing systems to indicate some sort of sound modification, such as voicing, lenition/fortition (turning a stop into a fricative or vice versa), vowel length, or tone.
To increase the character inventory
If a language borrows a script from another language, it might not have enough characters for all of its phonemes. One common option is to use diacritics to make more characters. This is common when languages adapt the Latin script, such as in French or Vietnamese.
Or because of language change, the writing system may no longer be adequate if new phonemes arise. One option is to add a mark to an existing character. (This is probably actually the origin of all of French's diacritics.)
To improve legibility
Diacritics can be used to improve legibility, which is the origin of the dot of "i" and "j". Some languages also use diacritics to distinguish homophones.
To mark syllables
The diaereses is used to indicate that two adjacent vowels are to be pronounced separately, as in the name Noël.
This is not a complete list. The Wikipedia page for diacritics gives a thorough look at what writing systems for conlangs use them for.