Essentials are:

-spare tube; ($5)
-tire irons; (cheap)
-pump or CO2 inflator ($20-35)
-patch kit; ($3)
-a few Allen wrenches to adjust "stuff";
-a seat or frame bag to stow it in ($15-35).

You will very soon want:

-bike shorts;
-clip-in pedals and shoes ($$)
-N+1 bike