Chemotrophs are organisms that feed on chemicals not produced by other living things. The most famous example is the bacteria that live near underwater hydrothermal vents. These vents constantly replenish the chemicals that the bacteria feed on. The vents don't provide the sort of chemicals an animal could eat; only microorganisms are able to do the chemistry necessary to obtain energy from them. However, the giant tube worms found near hydrothermal vents are animals that sustain themselves by hosting a symbiotic colony of such microorganisms.
The very first living things were probably chemotrophs - photosynthesis evolved later. (Fun fact: humans are causing a mass extinction, but we're not the first living things to do that. The honor goes to the early photosynthetic organisms which filled Earth's atmosphere with poisonous gas.)