How does the Mosher ester method allow determination of absolute configuration of secondary alcohols
The NMR spectra of two enantiomeric molecules are necessarily identical (just as all properties of enantiomers are identical in the absence of an external chiral environment). In order to tell the enantiomers apart, some additional chirality must be introduced. Two methods exist for this in NMR spectroscopy: 


Chiral solvating agents, such as lanthanide shift reagents
Chiral derivatising agents, such as Mosher esters 


In modern organic synthesis, the use of chiral derivatising agents is far more common, and the process can be summarised as follows: 



Outline of the Mosher ester analysis. Taken from Organic Spectroscopy and Structure Elucidation lecture notes (not available online) 

a. The alcohol of unknown configuration is derivatised by forming both the (R) and (S) Mosher esters. This can either be done under Steiglic conditions (DCC,DMAP) or via initial formation of the acid chloride. 

b. Proton NMR spectra are ran of the two diasteromeric esters. The analysis is often ran on the crude esters, therefore the acquisition of COSY/HSQC is also useful to aid in assignment. 

c. The difference between the chemical shifts of the (R) and (S) esters are tabulated (by convention we consider R-S, it makes no fundamental difference to the outcome however).  Note that the method does not consider the shift of the underivatised alcohol at all, however the peaks for the (R) and (S) ester must be unambiguously assigned. Note also that we do not take into account the oxymethine proton (the shifts of this are erratic and don't contribute to the analysis). 

d. The configuration is assigned by considering which side of the molecule is most effected by the presence of the Mosher ester. 

Step d is often the hardest to grasp conceptually. When we make the Mosher esters, what we're counting on is that they adopt a specific conformation (which they do). In this conformation, the phenyl group of the Mosher ester will shield part of the molecule (which is different depending on which enantiomer of the Mosher ester we're looking at). Consider the following example, in which the side shielded by the phenyl ring has a lower delta overall: 



Shielding by the Mosher ester. Taken from Harvard CHEM2016 lecture notes, Eugene Kwan 

The following conformational diagram is also (possibly more) instructive: 



Shielding by the Mosher ester. Source unknown 



The only way to actually understand the Mosher ester method is to carry it out (ideally with something of unknown stereochemistry so you can't convince yourself of the answer simply because you know its the case. 

The next best thing is a worked example: 



Outline of the Mosher ester analysis. Taken from Organic Spectroscopy and Structure Elucidation lecture notes (not available online) 

Looking at the two spectra, it becomes apparent that (R)-(S) is positive on the LHS methyl group, and negative on the protons on the RHS propyl group. Fitting this to the model, we can assign the sterocente of the secondary alcohol as (R). 



The reason both Mosher esters should be formed is due to the fact that change in chemical shift between one Mosher ester and the free alcohol is often not huge. Other derivitising agents (such as AMA) can be carried out with formation of a single ester.