What is a direct proof, formally? In my logic class I took in college, I sometimes would get wrong marks because I proved a statement indirectly instead of directly. Informally, a direct proof of a conditional $P \implies Q$, is one where you assume $P$ and try to deduce $Q$ "directly", without using the contrapositive of that conditional. But no one has ever given me a formal, rigorous definition of direct proof. Can someone give me one, and with one, can someone give examples of statements which can't be proven directly, only indirectly?

As Rob Arthan said in the comments, one way to distinguish between (one kind of) direct proof and indirect proof is simply to distinguish between proofs that use intuitionistically valid deductive rules and proofs that do not.
To show that there are some theorems (using classical logic) that have no intuitionistically valid proofs, it suffices to show that there is a way to interpret intuitionistic logic such that the intuitionistic deductive rules are sound for that interpretation, and also that there is a tautology of classical logic that is not a tautology under that interpretation. A popular one is LEM (i.e. $A∨¬A$), but it may not be convincing. Here is another:

$(A⇒B)∨(B⇒A)$.

You cannot prove this classical tautology in intuitionistic logic. In that precise sense there can be no direct proof of this tautology. This is more convincing than LEM because it is even slightly counter-intuitive and you really need to 'check cases' (relying on LEM) to prove it. Lest you think that 'checking cases' is direct, observe that every intuitionistic proof of ( $¬B⇒¬A$ ) can be translated to a proof that is intuitionistic except for a single LEM case-checking:
  If $A$:
    $B∨¬B$.
    If $B$:
      $B$.
    If $¬B$:
      [Insert proof of ( $¬B⇒¬A$ ) here.]
      $¬A$.
      Contradiction (i.e. $A,¬A$).
      $B$.  [by explosion]
    $B$.
  $A⇒B$.
There are many interpretations of intuitionistic logic under which this classical tautology does not hold. One is Kripke frames (see here for a sketch), and another is the BHK interpretation. Understanding either of them will give you a clear idea of how exactly an intuitionistic proof can be considered to be 'direct'.
