The types of presupposition are:
Existential presupposition: it is the assumption of the existence of the entities named by the speaker.
For example, when a speaker says "Tom's car is new", we can presuppose that Tom exists  and that he has a car.
Factive presupposition: it is the assumption that something is true due to the presence of some verbs such as "know" and "realize" and of phrases involving glad, for example. Thus, when a speaker says that she didn't realize someone was ill, we can presuppose that someone is ill. Also, when she says "I'm glad it's over", we can presuppose that it's over.
Lexical presupposition: it is the assumption that, in using one word, the speaker can act as if another meaning (word) will be understood. For instance:
    Andrew stopped running. (>>He used to run.)
    You are late again. (>> You were late before.)Â
In this case, the use of the expressions "stop" and "again" are taken to presuppose another       (unstated) concept.
Structural presupposition: it is the assumption associated with the use of certain words and phrases. For example, wh-question in English are conventionally interpreted with the presupposition that the information after the wh-form (e.g. when and where) is already known to be the case. Â
When did she travel to the USA? ( >> she traveled)
Where did you buy the book? (>> you bought the book)
