The molecular weight of an unknown material can be determined when this material is in a dilute "ideal gas" state.

An ideal gas is defined as one in which there are no intermolecular forces between the atoms or molecules. In such a gas, all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature.

An ideal gas (like any other thermodynamic system) can be characterized by three state variables: pressure (P), volume (V), and absolute temperature (T). For ideal gas the relation between them is expressed by the formula:
P V = N k T,
where N is the number of molecules in the gas, and k is a physical constant, known as Boltzmann constant.

From this formula it follows that equal volumes of gases at the same temperature and pressure contain the same number of molecules regardless of their chemical nature and physical properties - a principle first stated in 1811 by the Italian chemist Amedeo Avogadro.
The ideal gas law works extremely well at high temperature and low pressure. It offers a way to extract the number of gas molecules N confined inside a given volume V.

If we now measure the total mass of the sample M (by, for instance, condensing it from gas to liquid), we can extract the molecular mass M/N.

The vapor will be at equilibrium with the liquid state (what physicists call "coexistence") only at low temperature and/or high pressure which are outside the ideal gas regime. In other words, gas-liquid coexistence can be understood only if the molecular interactions are taken into account. Therefore, the attempt to use the ideal gas law to the vapor above a liquid would yield a somewhat inaccurate estimate for the molecular weight.

If you know the vapor density, then you know the mass per volume of the vapor. Now, this isn't quite enough to determine the molecular weight yet. (I am assuming that the overall exercise you are being asked to do is to determine the molecular weight of a liquid by using data from its gas phase - a "classic" chemistry lab experiment.) The vapor density can change depending on the pressure and temperature of the gas. If we assume that the gas can be treated as an ideal gas, then we can use the ideal gas law of:
PV = n R T, where P is the pressure, V is the volume of gas, n is the number of moles of gas, R is a constant, and T is the temperature. If we rearrange that equation, we get: n / V = RT / P.

Once you know the temperature and pressure, then everything on the right side of the equation is known. Now, since the number of moles can be determined by:n = mass / molecular weight, and we know that mass over volume is the density, then we can substitute all that back into the rearranged ideal gas equation to give:
(Vapor density) / M.W. = RT / P
or, rearranging again:
MW = (Vapor density * P) /(RT)

Now, if the gas can not be considered ideal, then the problem gets considerably more difficult as the non-idealities of the gas would need to be considered.