The word mindfulness means different things to different people, just like innovation. Four types of usages are common: (1) as a practice anchored in Buddhist texts (2) as a secular discipline with a loose to no connection to Buddhist texts/practices (3) as a clinical/therapeutic intervention for mental health issues such as depression; and (4) as a fitness tool like yoga also available on mobile apps and smartwatches. I have been a student of mindfulness of the second type (secular discipline with a loose connection with Buddhist practices) for the past twenty-five years and a teacher for the past eight years. My book “Mindfulness: connecting with the real you” came from this perspective.
As we look at Pali and Sanskrit words for mindfulness, we must note that these are not the only languages in which mindfulness can be traced. Chinese, Tibetan, Korean, Japanese, and perhaps other languages also offer rich sources of old Buddhist texts and would carry words that are translated as mindfulness. I am restricting the scope to Pali/Sanskrit due to my ease of understanding terms and texts in these languages.
The two Pali (Sanskrit) words we will look at are: Sati (Smriti) and Vipassana (Vipashyana)
Sati (Smriti): In 1881, T W Rhys Davids, an English scholar of the Pali language and founder of the Pali text society, translated the seventh of the Noble Eightfold path sammā-sati as “right mindfulness, the watchful, active mind” . Literally, the Pali word sati or Sanskrit smriti means memory, however, Rhys Davids knew the intended meaning in the Buddhist texts goes beyond just memory. He wrote, “In No. 7 (samma-sati) sati is literally 'memory' but is used with reference to the constantly repeated phrase 'mindful and thoughtful' (sato sampajâno); and means that activity of mind and constant presence of mind which is one of the duties most frequently inculcated on the good Buddhist."
Satipaṭṭhānā Suttā is one of the most used Buddhist texts as a source for mindfulness . It presents four establishments of mindfulness – cattaro satipaṭṭhānā .
Kāye kāyānupassī …vedanāsu vedānanupassi…citte cittānupassi…dhammesu dhammānupassi viharati ātāpi sampajāno satimā, vineyya loke abhijjhādomanassaṃ.
To live witnessing the reality of body as body…feelings as feelings…mind as mind…phenomena as phenomena, ardently, with clear comprehension, with awareness, keeping away from craving and aversion towards the world.
In this commonly occurring phrase, sati is translated as awareness. However, the word sati is always accompanied by anupassanā and sampajannā. Bhikkhu Bodhi translates them as observation and clear comprehension respectively while S N Goenka translates them as witnessing and wisdom of arising and passing respectively. In either case, when mindfulness is meant to capture the essence of Satipaṭṭhānā suttā, it makes sense for the word to carry the shades of sati, anupassanā and sampajannā.
The following verse from Dhammapada (Pali verse no. 374) also brings out the close relationship between sati and wisdom of arising and passing .
yato yato sammasati, khandhānaṁ udayabbayaṁ, labhatī pītipāmojjaṁ, amataṁ taṁ vijānataṁ
Whenever he sees with insight the rise and fall of the aggregates, he is full of joy and happiness. To the discerning one, this reflects the Deathless. (Translated by Acharya Buddharakkhita )
Vipassanā (Vipaśyanā): Mindfulness is also used as a translation of the Pali (Sanskrit) word Vipassana (Vipaśyanā). It literally means seeing clearly or seeing in a special way. It is also translated as insight. S. N. Goenka defines it as :
Paññatti ṭhapetvā visesena passati’ti vipassanā
Having removed the apparent truth, seeing things by their characteristics.
Vipassana represents a meditation tradition that got started in Burma (Myanmar) in the 18th century and spread among monks as well as laypeople in the 19th and 20th centuries in Burma, Thailand, and subsequently across the world . These traditions use both samaṭhā (tranquillity) and vipassanā (insight) by placing varying degrees of emphasis on the two aspects . However, many old Buddhist texts use these two words together. For example, in one of the discourses in Samyutta Nikāya (SN 43) Buddha says :
Katamo ca bhikkhave asaṅkhata-gāmī Maggo? Samatho ca vipassanā ca.
And what is the path that leads to the unconditioned? Tranquility & insight…
If the objective of Vipassana is to see clearly, we can ask what is it to be seen clearly? The answer differs somewhat based on the tradition. However, three characteristics, tilakkhanā (trilakśanā) are common, especially in Theravada tradition: namely anicca (impermanence), dukkha (unsatisfactoriness), and anatta (no self). For example, verses 277, 278, and 279 from Dhammapada express this:
sabbe saṅkhārā aniccā ti, yadā paññāya passati, atha nibbindatī dukkhe esa maggo visuddhiyā (277)
“All conditioned things are impermanent” - when one sees this with wisdom, one turns away from suffering. This is the path to purification. 
sabbe saṅkhārā dukkhā ti, yadā paññāya passati, atha nibbindatī dukkhe esa maggo visuddhiyā. (278)
“All conditioned things are unsatisfactory” - when one sees this with wisdom, one turns away from suffering. This is the path to purification.
sabbe dhammā anattā ti, yadā paññāya passati, atha nibbindatī dukkhe esa maggo visuddhiyā. (279)
“All things are not-self” - when one sees this with wisdom, one turns away from suffering. This is the path to purification.
For Nagarjuna, the father of Madhyamaka school, as he presents in MulamadhyamakaKarika verse 24.40, seeing just one characteristic that is of śūnyatā or essencelessness in everything is enough and everything else follows from it .
To summarize, we looked at two Pali/Sanskrit words – sati and vipassanā which have been translated as mindfulness. As we saw, in the Buddhist texts, these words are accompanied by other related words whose meanings are also incorporated in the meaning of mindfulness. Phrases like awareness of or seeing clearly into, with a calm mind, the characteristics of human existence such as impermanence, unsatisfactoriness, no-self, śūnyatā stand out.
 T W Rhys Davids, “Buddhist Suttas, translated from Pali”, Oxford at the Claredon Press, 1881, pg 107.
 T W Rhys Davids, pg 145
 Bhikkhu Bodhi, “What does mindfulness really mean?”, in “Mindfulness: diverse perspectives on its meaning, origins, and multiple applications at the intersection of science and dharma”, edited by J. Mark G. Williams and Jon Kabat-Zinn, 2015, pg 19.
 Achara S. N. Goenka, “Discourses on Satipatthana Sutta”, Vipassana Research Institute, 2013, pg 22.
 Anandjoti Bhikkhu, “A comparative edition of the Dhammapada”, 4th revised edition, April 2020, pg 221.
 Acharya Buddharakkhita (translator), “The Dhammapada: The Buddha’s path of wisdom”, Buddhist Publication Society, Kandy, Sri Lanka, 1985, pg 80.
 Achara S. N. Goenka, “Discourses on Satipatthana Sutta”, Vipassana Research Institute, 2013, pg 4.
 Patrick Pranke, “On saints and wizards: Ideals of human perfection and power in contemporary Burmese Buddhism”, Journal of International Association of Buddhist Studies, vol 33, 2010, pg 453-488.
 Bhikkhu Analayo, “The dynamics of Theravada insight meditation”, Zhuang Guobin (ed), Dharma Drum Publishing, Taiwan, 2012.
 Acharya Buddharakkhita (translator), “The Dhammapada: The Buddha’s path of wisdom”, Buddhist Publication Society, Kandy, Sri Lanka, 1985, pg 65 Nagarjuna’s sunyata through Mulamadhyamakakarika verses, October 2022