Here from the Sutta

The Blessed One said, "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." - sn22.48

This from abhidhamma

Therein what is faculty of mindfulness? That which is mindfulness, constant mindfulness, recollection, mindfulness, act of remembering, bearing in mind, non-superficiality, non-forgetfulness, mindfulness, faculty of mindfulness, power of mindfulness, right mindfulness. This is called faculty of mindfulness.

 

Mindful.’ Herein, what is mindfulness? That which is mindfulness, recollection, recall, mindfulness, remembrance, bearing (in mind), not losing, not confusing, mindfulness, the Faculty of Mindfulness, the Strength of Mindfulness, Right Mindfulness: this is called ‘mindfulness.’

 

With this mindfulness he is endowed, truly endowed, having attained, truly attained, being possessed, truly possessed, furnished (with it). Because of this ‘mindful’ is said. - vibhanga

Here from the Sutta

The Blessed One said, "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." - sn22.48

This from abhidhamma

Therein what is faculty of mindfulness? That which is mindfulness, constant mindfulness, recollection, mindfulness, act of remembering, bearing in mind, non-superficiality, non-forgetfulness, mindfulness, faculty of mindfulness, power of mindfulness, right mindfulness. This is called faculty of mindfulness.

Mindful.’ Herein, what is mindfulness? That which is mindfulness, recollection, recall, mindfulness, remembrance, bearing (in mind), not losing, not confusing, mindfulness, the Faculty of Mindfulness, the Strength of Mindfulness, Right Mindfulness: this is called ‘mindfulness.’

With this mindfulness he is endowed, truly endowed, having attained, truly attained, being possessed, truly possessed, furnished (with it). Because of this ‘mindful’ is said. - vibhanga