Short answers, pulled from the story.
Satipatthana is a Pali compound translated as the establishment of mindfulness, the presence of mindfulness, or the foundations of mindfulness. Scholars such as Bhikkhu Analayo and Bhikkhu Bodhi favor presence or establishment, based on how rarely the term behind foundation appears in the earliest texts, while the translator Soma used foundations of mindfulness.
Satipatthana practice directs mindfulness toward four domains: the body, feelings, the mind, and dhammas, meaning principles or categories of the Buddha's teaching. Rupert Gethin translates the shared early formula as an instruction to dwell watching each of these in turn, with ardency, clear comprehension, and freedom from desire and discontent.
The Satipatthana Sutta is probably the most influential meditation text in modern Theravada Buddhism, and it underlies the teachings of the Vipassana movement. It appears in the Majjhima Nikaya as MN 10 and, with an added section on the four noble truths, in the Digha Nikaya as DN 22.
Early texts such as MN 44.12 state that the four satipatthanas are the basis for samādhi, and Bhikkhu Sujato concludes that satipatthana's main role is to support jhana. Thanissaro Bhikkhu argues from suttas including MN 125 and AN 8:63 that satipatthana practice culminates in reaching the first jhana.
The five hindrances named in the Satipatthana Sutta are sense desire, ill will, sloth-and-torpor, restlessness-and-worry, and doubt. Early discourses describe removing them as necessary for attaining jhana, since the hindrances are said to choke the mind and rob understanding of its strength.
Many modern Theravada and Vipassana teachers treat Satipatthana as centered on moment-to-moment awareness of rising and falling mind-body phenomena, a reading Bhikkhu Sujato calls the vipassanāvāda. Other teachers, including those of the Thai forest tradition, reject a strict split between vipassana and samatha and treat the two as complementary.