What is the conversion factor relating volume and moles called? Write the relationship or the conversion factor.?

Remember there are three equations relating to moles. they are:- 

moles(solids) = mass(g) / Mr 

moles (liquids)= [conc] x vol(mL) / 1000

moles(gases) = PV/RT

These three equations are mathematically interchangeable.  e.g. 

mass(g)/Mr = PV/RT 

However, the conversion fctor for liquids is 

moles = [conc] x vol(mL) / 1000

NB The '1000' is needed to convert 'mL' to 'L/dm^3' because conc'n is quoted in 'moles dm^-3' or 'mole L^-1'. 

NB 'dm^3 and 'L' are the same volume with different names.  

As previous responders answered, it's the molar volume; for gases, it's calculated using the ideal gas equation:

PV = nRT

Rearrange that to:

n/V = P/(RT)

(P, V, T, and R are in appropriate units to give moles/L)

For liquids and solids, it's calculated from the density (p [Greek rho], in g/mL)) and molecular (or formula) weight (MW, in g/mole) for a given substance:

Molar Volume (mole/L) = (p/(MW)) x (1000 mL/L)

Ideal Gas Law: PV=nRT

n is moles, R is gas constant (use the correct one!),

T is the temp in K (C+273.15)

V is the volume in Litres

you'll need to be more specific

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If we're talking about the volume of a gas, the requested relationship is called the molar volume.

At 1 atm and 0°C,

22.4 L = 1 mole

[If we're talking about liquids or solids there is no direct conversion factor between  volume and moles.  The conversion can be made, but it is a multi-step process which requires more than one factor.]

There isn't any. Mole is a number , like million is

How many moles of molecules fit in a liter depends on the density of the substance, aka the size and shape of the molecules