One newton-metre is 8.8507457913 pound-inches. The pound-inch is the small-torque unit of American practice, used where the pound-foot would give an awkward fraction — and where over-tightening does the most damage.
About the units
A pound-inch, written lbf·in, is one pound-force acting one inch from the axis. It is one twelfth of a pound-foot, following the same twelve that relates the length units. American torque specifications switch to pound-inches below roughly ten pound-feet, because 3.5 lbf·ft is less legible than 42 lbf·in and because torque wrenches for small fasteners are calibrated in the smaller unit. Metric practice does not make the same split: newton-metres cover the whole range, with decimals where needed.
The exact factor
The exact value is 1 lbf·in = 0.112984829027617 N·m, so a newton-metre is 8.85074579132718 pound-inches. Useful anchors: 1 N·m is 8.85 lbf·in, 5 N·m is 44.3, and 10 N·m is 88.5. The relationship to pound-feet is exactly twelve, so a figure in pound-inches divided by twelve gives pound-feet. Small electronics and optical assemblies are specified in the range of 2 to 20 lbf·in, which is 0.23 to 2.3 N·m — precisely the range where a full-size torque wrench cannot be trusted.
Where you meet this conversion
Small fasteners in electronics, aerospace and precision assembly, where thread damage is the main risk and specifications are tight. Aircraft maintenance manuals use pound-inches extensively for instrument and avionics mounting. Bicycle components — stem bolts, seat clamps, carbon fibre parts — are specified in newton-metres in Europe and pound-inches in America, and the consequence of over-tightening a carbon component is a crack rather than a loose bolt. Firearm scope mounts and optical instrument assembly use the same range.