Right Mindfulness is the seventh of the eight path factors in the Noble Eightfold Path, and belongs to the concentration division of the path. 

The definition (the four frames of reference)

"And what is right mindfulness?  There is the case where a monk remains focused on the body in & of itself — ardent, alert, & mindful — putting aside greed & distress with reference to the world.  He remains focused on feelings in & of themselves... the mind in & of itself... mental qualities in & of themselves — ardent, alert, & mindful — putting aside greed & distress with reference to the world.  This is called right mindfulness... "This is the direct path for the purification of beings, for the overcoming of sorrow & lamentation, for the disappearance of pain & distress, for the attainment of the right method, & for the realization of Unbinding — in other words, the four frames of reference." —  DN 22

Abandoning the wrong factors of the path

"One is mindful to abandon wrong view & to enter & remain in right view: This is one's right mindfulness... "One is mindful to abandon wrong resolve & to enter & remain in right resolve: This is one's right mindfulness... "One is mindful to abandon wrong speech & to enter & remain in right speech: This is one's right mindfulness... "One is mindful to abandon wrong action & to enter & remain in right action: This is one's right mindfulness... "One is mindful to abandon wrong livelihood & to enter & remain in right livelihood: This is one's right mindfulness..." — MN 117

Like balancing a pot of oil on one's head

"Suppose, monks, that a large crowd of people comes thronging together, saying, 'The beauty queen!  The beauty queen!'  And suppose that the beauty queen is highly accomplished at singing & dancing, so that an even greater crowd comes thronging, saying, 'The beauty queen is singing!  The beauty queen is dancing!'  Then a man comes along, desiring life & shrinking from death, desiring pleasure & abhorring pain.  They say to him, 'Now look here, mister.  You must take this bowl filled to the brim with oil and carry it on your head in between the great crowd & the beauty queen.  A man with a raised sword will follow right behind you, and wherever you spill even a drop of oil, right there will he cut off your head.'  Now what do you think, monks: Will that man, not paying attention to the bowl of oil, let himself get distracted outside?" " No, lord." " I have given you this parable to convey a meaning.  The meaning is this: The bowl filled to the brim with oil stands for mindfulness immersed in the body.  Thus you should train yourselves: 'We will develop mindfulness immersed in the body.  We will pursue it, hand it the reins and take it as a basis, give it a grounding, steady it, consolidate it, and undertake it well.'  That is how you should train yourselves." —  SN 47.20

Meditation on death

"Mindfulness of death, when developed & pursued, is of great fruit & great benefit.  It plunges into the Deathless, has the Deathless as its final end.  Therefore you should develop mindfulness of death." —  AN 6.19

Meditation on breathing