Reaction(s) involved in thermite process can be
Goldschmidt thermite process: It involves a highly exothermic reaction involving a fuel (like powdered aluminium) and a metal oxide. At high temperatures, aluminium reduces these metal oxides to the corresponding metals:
3TiO2(s) +4Al(s) → 2Al2O3 (s) +3Ti(l)
Cr2O3 (s)+2Al(s) → Al2O3 (s)+2Cr(l)
3Mn3O4+8Al→9Mn+4Al2O3
Do note that the reduced metal is in the fused form.
When Fe2O3 is used, this process is also known as thermite welding (used to join railway tracks). But this is not the preferred method for extraction of Iron commercially.
2Fe+Al2O3→2Al+Fe2O3