So an 8' long board 12" wide and 1" thick is 8 board feet. Or half as wide and twice as thick (6" x 2") in the same length is also 8 board feet. You get the idea.
Board thicknesses are measured in the "quarter system" - 4/4 (pronounced "4 quarter", that is 4 quarters of an inch) is a 1" thick rough-rough sawn board; 8/4 (8 quarter) is a 2" thick rough board. As this is rough sawn (super rough and unsightly finish), you usually lose at least 1/8" while finishing the board, so a 4/4 rough board gives you something around 3/4" finished product.
This leads to 5/4 being another fairly common dimension, as you can get a 1" finished board out of it. 6/4 and 12/4 are somewhat common as well.
It might seem like an "ugly dimension" compared to something like m^3, but its really built around speed. Watching someone skilled price out an enormous stack of mismatched lumber (width, thickness, and length) in seconds with a lumber rule is always a fascinating process to watch (https://www.popularwoodworking.com/techniques/the-lumber-rul...).
Usually the $/board foot increases slightly at higher dimensions as thicker lumber is harder to produce. I got some 4/4 black walnut for $12.80/bdft last week; 8/4 was closer to $16.
Edit: This is also why 2x4s are actually 1.5"x3.5" - historically they were 2x4 rough sawn lumber, but .5" is lost in both dimension when producing finished boards you purchase in the store.
I imagine there are people all over the world doing the same tricks with metric measurements.
To get a 1" finished board you need to start with at least 5/4, because you may have some twist in the board and you can flatten alternating sides to make it flat. If you were just running it through a machine without paying attention, your might need to take that same board down to 0.75" to get it flat.
But they aren't wasting time like that for framing lumber. But when your lumber is expensive, it's worth a little focused labored to use more of the board
In the hardwoods I usually work with (walnut/maple/padauk) its fairly minimal. A nominally 4/4 board usually comes in about 1/16" small; which is a lot less than I lose running it through the planar. Some of that could be the sawmill skimping out too and cutting just a bit undersized....
I'm finding quoted numbers between 2-8% volumetric shrinkage (https://www.wood-database.com/wood-articles/dimensional-shri...), which is still quite a bit less than you find in the small dimension of a 2x4 (2"-> 1.5%, or ~33%)
Dimensional lumber is the original shrinkflator
So in a 10ft long 2x4 it would be the actual size of a 2x4 (1.5"x3.5"x10').
A standard board foot would be (1"x12"x12"), or any other math that makes up the same volume.
Just like with board feet it doesn't say anything about the actual dimensions. Also, I don't think it's much better as a unit than board feet either, both have very unclear relationship to the actual amounts you need. You'll always need to do a pretty complex calculation to figure out how much you'd need to buy. The pricing is mostly useful as a comparison to look at relative or day to day pricing differences.
A board foot is a measure of volume - with an implicit third dimension.
So you could just use the SI unit for volume.
A board foot ~= 2.4 litres.
This seems a bit silly though, as timber is not fungible in the way you'd expect a volume of something to be.
As noted by sibling comments, you very much can buy timber in cubic metres - a convenience for costing / transport - but you're not going to buy timber without knowing dimensions of the individual lengths.