Concept of Vedana as expounded in Theravada Buddhism Throughout canonical discourses (Sutta Pitaka) the Buddha teaches that there are 3 modes of Vedana.

Where in Pali canon do we find 6 classes of Vedana?

How is Vedana related to Nibbana?
Vēdanā is conventionally translated as “feelings”, but it is much more than just feelings. The Buddha said that vēdanā can be analyzed in many ways, with simple or deeper meanings. Though you have been told that there are six types of Vēdanā  there are actually 9 types of vēdanā.

A vēdanā can be a dukha vēdanā (due to a past bad kamma), a sukha vēdanā (due to a past good kamma), or a neutral sensation — like feeling the wind on the body — called an upekkha vēdanā. However, normal humans go far beyond that and generate three additional types of their own mind-made “samphassa ja vēdanā”. 

A normal human being will mentally generate three more type of vēdanā due to those sukha, dukha, and neutral vēdanā that initially arise due to kamma vipāka via all five physical senses. If it is dukha vēdanā (due to a headache, getting injured, etc. ), one is likely to start worrying about it and add more suffering. Those are dōmanassa vēdanā or āmisa dukha vēdanā.

If it is sukha vēdanā (getting a massage, lying in a luxurious bed, etc. ), one is likely to start generating thoughts about how good it is and how one can enjoy similar sukha vēdanā in the future. Those are sōmanassa vēdanā or āmisa sukha vēdanā ; one could also be generating them by remembering past such sukha vēdanā. These are also added in by the mind. Sōmanassa means “mind-made joyful”. Dōmanassa means “mind-made misery”, which is basically a depressed mindset.

When one stays away from generating too many “samphassa jā vēdanā” of both kinds, one will start feeling three more types of vēdanā. These are called nirāmisa vēdanā, because they arise due to staying away from cravings (and repulsion) to worldly objects. When one prevents the mind from heating up by comprehending the adverse effects of “san” and staying away from them, one’s mind starts “cooling down”. This is the nirāmisa sukha vēdanā. This is what is emphasized by “ätäpi sampajäno” in the Mahā Satipatthāna Sutta; it means “remove the fire or heat from one’s mind by being aware of the ‘san‘ or “immoral tendencies”. 

Then there are vēdanā that one experiences when proceeding on the Noble Eightfold Path. Sometimes, one gets a bit discouraged by not advancing “fast enough” on the Path.  Those are not dōmanassa vēdanā (because they are devoid of patigha anusaya); they are nirāmisa dukha vēdanā. 

The Sutta references are many, and I will refer you to one – the Ananda Sutta: Leading to Awakening (SAMYUTTA NIKAYA 12:10). 

How vēdanā is related to Nibbana will call for a lengthy explanation, and you will not understand it if explained it in a paragraph or two. One day you will get to understand it proper, if you try to make sense of the following: 
“Sukha and dukha vedana arise due to kamma vipaka. Somanassa and domanassa vedana arise due to sankhara, which in turn arise due to our gathi and asavas. The more sankhara we do, the stronger a given gathi (habit) becomes, which in turn become asavas (cravings) and fuel the sansaric journey (rebirth process). This vicious cycle can be broken only through comprehending the anicca nature of this world.”