See here: Complete definition:

Hypergeometric2F1[a, -n, -m, z] == 
Sum[((Pochhammer[a, k] Pochhammer[-n, k])/(Pochhammer[-m, k] k!)) z^
 k, {k, 0, n}] /; 
Element[m, Integers] && m >= 0 && Element[n, Integers] && n >= 0 && 
 m >= n

Then for your example:

$$\, _2F_1(a,-3;-3;z)=\sum _{k=0}^3 \frac{\left((a)_k (-3)_k\right) z^k}{(-3)_k k!}=1+a z+\frac{1}{2} a (1+a) z^2+\frac{1}{6} a (1+a) (2+a) z^3$$

 Hypergeometric2F1[a, -3, -3, z] // Expand
 (*1 + a z + (a z^2)/2 + (a^2 z^2)/2 + (a z^3)/3 + (a^2 z^3)/2 + (
  a^3 z^3)/6*)
 Sum[((Pochhammer[a, k] Pochhammer[-3, k])/(Pochhammer[-3, k] k!)) z^
     k, {k, 0, 3}] // Expand
  (*1 + a z + (a z^2)/2 + (a^2 z^2)/2 + (a z^3)/3 + (a^2 z^3)/2 + (
   a^3 z^3)/6*)

Regards M.I.