If an alien species ate minerals (aka a lithovore species), how might their bodies differ from ours? How would they eat/digest the materials? Do those answsers change if they eat substances like diamond versus metals like gold?
Animals like us eat for two reasons: to get carbon that we burn for energy, and also carbon and nitrogen (protein) that we turn into our tissues. Bacteria have those same two needs but sometimes have the two functions separate. They might get energy via oxidizing or reducing inorganic chemicals / minerals - even gold. They need carbon just like us but might be able to use oxidized carbon (CO2) or reduced carbons we cannot use (for example alkane petrochemicals).
Eating 101. 1: Break it up so you can take it in and work with it. This can be mechanical (teeth, gizzards) or enzymatic or both. Enzymes can work inside the body (as we do) or outside (as fungi do - they squirt the stuff around and then sop up the breakdown products). Also, breaking it up increases surface area which facilitates step 2: chemistry.
2: Work chemistry on it. We oxidize the food molecules we make into energy, carefully, via enzymes which is a narrower term for a catalyst. Any kind of chemistry that is energetically favorable in your circumstance can be harnessed to produce energy for the organism. If you want to use materials you ingest to make your own tissues, then you work that kind of chemistry to make them into you.
If we had commensal organisms able to oxidize more exotic carbon sources then we could use those carbon sources for food. That works for cows and termites. If we had commensal organisms capable of metabolizing sulfur into energy then we could chew up sulfur-containing compounds and present them to our commensals for processing. That works for some types of clam and worm.
The same way it works for plants: picture the gut uses the processes used by plant roots to extract needed types of atoms from minerals.
The animal still needs energy from someplace! If it uses sunlight or chemical energy like some bacteria, it will have a slow metabolism and be more like coral than what you normally think of as animals.