Mindfulness is a preverbal (pre-thinking) mental quality that remembers to be present. It is the bare, choiceless, moment to moment non-comparing and non-judging attention to the mental or physical activity that is occurring here and now. It pays equal respect to the pleasant and unpleasant objects. It also possesses a quality of understanding, patience, and acceptance toward all that is occurring in the present moment.
Mindfulness is one of the "universal" wholesome (beautiful) mental factors that when fully present, will enhance other wholesome mental qualities and suppress the unwholesome ones, such as anger, aversion, fear etc... Therefore praciticing mindfulness is a way to make one mind beautiful.

There are four ways (or aspects) of establishing mindfulness:

1) Body or physical aspect (Kayanupassana):

-Body postures (sitting, standing, walking, lying)

-Movements

-Physical sensations within the body

-The breath.

A direct way to experience physical sensations is, rather than the concept of “my body” with its form or shape, to be aware of the texture (hard or soft, rough or smooth, light or heavy), its temperature (warm or cool), its dynamics (motion, vibration, or tension/pressure) and its’ cohesiveness or fluidity (which is generally too subtle to directly experience).  These four kinds of manifestation are traditionally known as the earth, fire, air (wind), and water elements (pathavi-dhatu, tejo-dhatu, vayo-dhatu and apo-dhatu,).

2) Feeling (Vedananupassana): Mindfulness of pleasant, unpleasant, or neutral feeling tone of a physical or mental object (the current activity that one is aware of).

3) Mind or consciousness/mental states (Cittanupassana).

4) Mental objects/contents/phenomenas (Dhammanupassana). 

There are overlaps between mind and mental objects therefore, to simplify, 3) and 4) could be consider together as mindfulness of the mind. This involves awareness of thoughts (thinking, reflecting, remembering, planning etc…), mental states including emotions (sadness/joy, fear/hope, aversion/appreciation, anger/love, confusion/clarity, drowsiness/agitation etc…) or consciousness itself (the knowing mind). It also includes the observation of specific mental qualities such as the hindrances (or difficulties)  of the practice,  or awakening (insight/enlightenment) factors and the awareness of objects of the sense doors such as seeing, hearing, tasting, smelling, touching etc…, including the mind's reaction to them. We can also see how we are caught in these experiences, thus free ourselves from them.

In practice, one does not need to figure out which element or foundation the object represents but simply to be aware of them.  It is helpful however to know which experiences are reality (see Concept and reality) and pay more attention to them.