A board foot is a volume, 144 cubic inches: a piece one foot long, one foot wide and one inch thick. Softwood dimension lumber is counted on its nominal size, the size before drying and planing, so a 2x4 counts as 2" × 4" even though it measures 1-1/2" × 3-1/2". The takeoff calculator adds up board feet line by line and the chart lists the real sizes you will be cutting.
The formula
Multiply nominal thickness by nominal width (inches) by length, then divide by 144 if the length is in inches or by 12 if it is in feet. A 2x12 × 8 ft is 2 × 12 × 8 ÷ 12 = 16 board feet; a 2x4 × 8 ft is 5-1/3.
BF = T × W × L(in) ÷ 144 = T × W × L(ft) ÷ 12 (nominal inches)
Why a 2x4 is not 2 by 4
Sawmills cut green boards close to the nominal size, then the wood shrinks as it dries and loses more when it is planed smooth. The American Softwood Lumber Standard (PS 20) fixes the dry dressed sizes: 1x boards are 3/4" thick, 2x lumber 1-1/2", 4x 3-1/2". Widths up to 6" lose 1/2" (a 2x6 is 5-1/2"), and 8" and wider lose 3/4" (a 2x8 is 7-1/4", a 2x12 11-1/4").
5/4 decking is the odd one: it is counted as 1-1/4" but measures 1" thick, which is why the stair calculator takes 1" off the bottom riser for 5/4 treads.
Hardwoods use the quarter system
Hardwood is sold rough by the board foot in quarters of an inch: 4/4 is 1" rough, 8/4 is 2". Board feet use the rough thickness, and widths are random, so measure each board and add them up.
Waste and ordering
Allow 10% for framing and 15% for decking with angled ends or picture-frame borders. Order lengths that cut cleanly: three 36" treads come out of a 10 ft board with room for the kerfs and end checks, four out of a 12.
Many yards price dimension lumber per piece or per linear foot rather than per board foot. The takeoff converts any line to board feet so you can compare quotes on the same basis.
A worked example: one deck stair
Four 2x12 × 8 ft stringers are 4 × 16 = 64 board feet. Four 5/4 × 6 × 12 ft boards for the treads are 7.5 board feet each, 30 in all, and two 4x4 × 8 ft posts add 21-1/3. With 10% waste the stair is about 127 board feet, which is the number to compare when one yard quotes per piece and another per board foot.