Why is Cr2O3 amphoteric but CrO not?
Basic oxide are those oxides that dissolve in water to form soluble hydroxide. Basic oxides are very soluble for this reason. Acidic oxides are that oxides that dissolve in water to form strong acids. Generally, covalent oxides form acidic oxides as the element to which oxygen is bonded is electronegative. As charge increases, electronegativity increase and thus oxide become acidic. This also applies for elements having multiple oxides. As oxidation no. increases, charge increases and thus oxides become acidic. This link shares a good information on this topic:


  Since the acidity of a cation rises rapidly with charge, d-block
  elements which exhibit a wide variety of oxidation numbers may have
  one or more oxides that exhibit only basic properties and one or more
  oxides that exhibit only acidic properties. The higher the oxidation
  number the more acidic the corresponding oxide. Chromium is an example
  of such an element. 
  
  $$
\begin{array}{|c|c|c|}\hline
\text{Oxide}&amp;\text{Oxidation number}&amp;\text{Category}\\\hline
\ce{CrO}&amp;\ce{Cr^2+}&amp;\text{basic}\\\hline
\ce{Cr2O3}&amp;\ce{Cr^3+}&amp;\text{amphoteric}\\\hline
\ce{CrO3}&amp;\ce{Cr^6+}&amp;\text{acidic}\\\hline
\end{array}
$$


The amphoteric character of chromium(III) is stated in its Wikipedia article:


  Chromium(III) oxide is amphoteric. Although insoluble in water, it
  dissolves in acid to produce hydrated chromium
  ions, $\ce{[Cr(H2O)6]^3+}$ which react with base to give salts
  of $\ce{[Cr(OH)6]^{3−}}$. It dissolves in concentrated alkali to yield
  chromite ions $(\ce{[CrO2]^{-}})$.


So, the rule of thumb is as oxidation number increases, charge on metal increases, acidic character increases. Also, basic oxide is soluble in water, acidic oxide is insoluble in water. This can be used to differentiate between acidic and basic oxide.