FINAL ANSWER: 1 cubic foot of air at standard temperature and pressure assuming average composition weighs approximately 0.0807 lbs.
This one doesn't have an answer that is exact or necessarily correct as air is a composite mixture of various gases and this mixture can change due to an enormous quantity of variables. Discussing these variables and how these variables affect the air is one of the foundations of meteorology - the study of weather.
But given some data and making a few assumptions we can come to an answer that is probably pretty close.
First of all, we need to know the composition of air. What is air made of and how much is it made of. A quick internet search popped up http://www.engineeringtoolbox.com/8_212.html which gives us something to work with. (For those unfamiliar with the site's punctuation, a comma "," is used instead of a period "." a decimal delineation mark). This breakdown is actually greatly affected by where we are in the world and how far we are above sea level and other factors. We will assume average composition at sea level at 0 degrees Celsius which is known as STP or Standard Temperature and Pressure.
For reasons that should become clear a bit later, we need the volume of the various components, not the weight. This is the composition of air (as cited from the website above) by volume:
Oxygen: 20.99%
Nitrogen: 78.03%
Carbon Dioxide: 0.03%
Hydrogen: 0.00005%
Argon: 0.93%
Neon: 0.0018%
Helium: 0.0005%
Krypton: 0.0001%
Xenon: 0.000009%
There is a neat property that applies to any gas that a 22.4 liters of it at STP (standard temperature and pressure) contains 6.02214199 x10^23 molecules of said gas. This amount (Avogadro's number) is called a mole. So another way to say this is that a singe mole of gas at STP will fill 22.4 liters.
The special thing about a mole is that a if we have a molecule, we can find that molecule's molecular weight (from a periodic table) and that's how many grams a mole of the molecule would weigh. For instance, if we were to look up iron (Fe) on a periodic table, we find that its atomic weight is 55.845. A mole of iron is therefore 55.845 grams.
Quick conversion of 1 cubic foot to liters gives us 28.32 liters. Going back we can take this number to find that 28.32 liters of gas at STP is going to be 1.26414494 moles of air. Great, we can now determine how many moles of each particular substance we have from the percentage composition of air. These are the results (and the molecular symbol for each):
Oxygen (O2): 0.2653 moles
Nitrogen (N2): 0.9864 moles
Carbon Dioxide (CO2): 0.0003792 moles
Hydrogen (H2): 0.0000006321 moles
Argon (Ar): 0.01176 moles
Neon (Ne): 0.00002275 moles
Helium (He): 0.000006321 moles
Krypton (Kr): 0.000001264 moles
Xenon (Xe): 0.0000001138 moles
By looking on the periodic table for the atomic weight of constituent atoms in the above molecules (for instance CO2 we would get the weight for C and add to it twice the weight of O) and multiplying by the number of moles we have of these substances, we will get the following weights for the substances:
Note: After rounding errors and such, our current accuracy is such that anything less than 0.01g is insignificant and I have left those numbers out.
Oxygen: 8.49g
Nitrogen: 27.63g
Carbon Dioxide: 0.02g
Argon: 0.47g
Adding these numbers together (remember the rest was insignificant) we get 36.61g. Converting to pounds (since the question was in English units I assume the answer is desired in English units) we get 0.0807 lbs.
FINAL ANSWER: 1 cubic foot of air at standard temperature and pressure assuming average composition weighs approximately 0.0807 lbs.
Answered by:
Richard Remer

'To myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.'