One pound-foot is exactly 12 pound-inches. The twelve comes straight from the twelve inches in a foot, and the pair exists because American torque specifications switch between the two units at around ten pound-feet.
About the units
Both are imperial torque units: a pound-force acting at a perpendicular distance, in one case a foot and in the other an inch. The relationship is therefore exactly the length relationship. American practice uses pound-feet for anything substantial — wheel nuts, cylinder heads, suspension bolts — and pound-inches for small fasteners where the larger unit would give fractions. The switchover is a matter of convention rather than rule, and some manuals give both.
The exact factor
The factor is exactly 12. Both directions are exact, and the arithmetic is easy enough to do in the head: 25 lbf·ft is 300 lbf·in, and 84 lbf·in is 7 lbf·ft. In metric, 1 lbf·ft is 1.3558 N·m and 1 lbf·in is 0.11298 N·m. The error worth guarding against is a torque wrench set to the wrong unit — a wrench marked in pound-inches set to 90 when the specification meant 90 pound-feet applies one twelfth of the intended torque, and the joint comes loose.
Where you meet this conversion
American automotive and aerospace maintenance, where a single service manual routinely uses both units within one procedure: engine mounting bolts in pound-feet, sensor and cover screws in pound-inches. Torque wrenches are sold in overlapping ranges, and choosing the right tool means knowing which unit the specification uses. Aircraft maintenance is particularly strict about this, since fastener torque is a recorded and inspected item.