mTOR (mechanistic target of rapamycin, also written FRAP1 in older literature) is a serine/threonine protein kinase — an enzyme that turns other proteins on or off by adding phosphate groups to them. It sits at the center of how a cell decides whether to grow: when nutrients, growth factors, and energy are abundant, mTOR is active and pushes the cell to build proteins and grow; when any of those are scarce, mTOR switches off and the cell shifts to conserving and recycling resources instead.
mTOR wasn't discovered by studying growth signaling directly — it was found because of a drug.
The mammalian version of TOR — what we now call mTOR — was cloned by three independent labs in 1994, using rapamycin itself as the molecular hook to fish it out.
mTOR works as the catalytic core of two distinct protein complexes with different jobs — mTORC1 and mTORC2, covered in full on their own answer page. In short: mTORC1 is the nutrient/growth-factor sensor that controls protein synthesis, autophagy, and growth — and is rapamycin's direct target. mTORC2 is involved in cell survival and glucose metabolism, and is only reached by rapamycin indirectly, with chronic dosing.
Because mTOR sits at the intersection of nutrient sensing, growth, and metabolism, its dysregulation shows up across an unusually wide range of conditions: cancer (see how mTOR connects to cancer), the genetic disorder tuberous sclerosis complex (where a mutation leaves mTORC1 stuck permanently "on"), and the biology of aging, where inhibiting it with rapamycin is the single most consistent way known to extend lifespan in laboratory animals across species.