Your example is tricky to handle because an answer depends upon where you're starting, i.e., what you may assume as known.Thus there are (at least) three approaches.

1. As noted in Sean Clarke's answer, you could directly calculate the contour integral from its definition. Parameterize the unit circle C:

   \[Gamma][t_] := Exp[2 \[Pi] I t]
   f[z_] := 1/z
   Integrate[f[\[Gamma][t]] \[Gamma]'[t], {t, 0, 1}]
   

2. If you already know Cauchy's Integral Formula, apply it directly.

3. Calculate the residue at the origin and use the Residue Theorem. The catch is — how do you find the residue? Mathematica will tell it to you:

    Residue[f[z],{z,0}]
   

But then how does Mathematica know that? You could further ask Mathematica for (part of) the Laurent expansion around the origin:

     Series[f[z],{z,0,3}]

But that rather begs the question as to how that expansion is determined.

The simplest way to know the residue at the origin is to note that, by definition, the function has a singularity inside C only the simple pole at the origin, so that Res(f,0) = lim_{z→ 0} z (1/z) = 1.