Why is AlCl₃ is known as aluminium chloride while PCl₃ is known as phosphorus trichloride?
Names of binary molecular compounds (which contain only two different elements, normally nonmetals, and do not contain positive and negative ions) can contain the prefixes mono, di, tri, tetra, penta, etc. 
Normally, names of ionic compounds do not use these prefixes to indicate the number of positive or negative ions. $\ce{CaCl2}$ is calcium chloride, not "calcium dichloride". 

However, aluminum chloride, $\ce{AlCl3}$, is sometimes called aluminum trichloride which is not incorrect in this case because it actually is a molecular compound (it has very polar aluminum-chlorine covalent bonds) even though it looks like it should be ionic since it contains metal and nonmetal elements typical of ionic compounds.
 $\ce{MnO2}$ is often called manganese dioxide rather than manganese(IV) oxide for the same reason. Usually, it is best to play it safe with compounds like these and use the ionic names:

$\ce{FeCl3}$ is ferric chloride or iron(III) chloride rather than "iron trichloride." $\ce{PCl3}$, a molecular compound (expected since only nonmetallic elements are present), is commonly called phosphorus trichloride, but is sometimes called "phosphorus(III) chloride."