What are "Supernatural mental abilities" in relation to Buddhism? In this question How to help non-Buddhists understand a young person's desire to be ordained as a monk?, one of the "selling points" listed was:


  5) Supernatural mental abilities (astral travel, reading minds, remembering past lives, etc.)


Could someone explain this is more detail? I had thought this was more of a pop culture point rather than what seems to be a core concept.  

Do such "abilities" actually exist, and if so what is their implication in the practice of Buddhism or proceeding in one's daily life? Do monks or others actively practice them?
The question is slightly broad and asks about several different facets.  I'm going to focus on "Do monks or others actively practice them?"  The answer to this is "yes, some do."

There are a variety of beliefs and practices in a wide array of different Buddhist traditions which could be considered to be "Supernatural" in some way, shape, or form. They could also be considered wholly "natural" depending on your point of view of the mechanics behind what is happening. 

There are practices such as tummo which have received some amount of study that allow for increasing the core body temperature.  This may or may not count as "supernatural" but it would certainly seem that way to an outside observer with no experience in the practice. 

There is a belief among some practitioners of certain sects that there is a literal rainbow body phenomena wherein the body releases light and, sometimes, shrinks (my source on this is a conversation with a monk, if someone knows of something written down I'd be interested).  It is believed that this can be attained through the practice of dzogchen.

Information on belief in various forms of an "energy body" or "subtle body" are everywhere in various Buddhist traditions. 

There's a practice called dream yoga which includes something that could very well be considered "astral travel," though it is notoriously difficult to take a classification system derived from one system and apply it to another, so whether you count it as "astral," "illusory," or "supernatural" or just something related to what others call that may be tricky. These practices certainly have been used historically for divination (as described by Tenzin Wangyal Rinpoche), among other purposes. 

This isn't even getting into the phenomena of emanations and other such beliefs.

Basically: There are certainly beliefs and practices, engaged in by both monks and non-monks, that can be interpreted this way.