Respiration uses an electron-transport chain to transfer electrons from reduced donors to a terminal electron acceptor while conserving part of the redox free-energy change in an ion gradient.

When molecular oxygen is the terminal acceptor, the process is aerobic respiration.

When a respiratory electron-transport chain instead uses another terminal acceptor, the process is anaerobic respiration.

Possible terminal acceptors in different microorganisms include species such as nitrate, sulfate or carbon dioxide. The exact products and energy yields depend on the organisms and pathways involved.

Anaerobic respiration is not fermentation

Both can operate without molecular oxygen, but their energy-conservation mechanisms differ.

anaerobic respiration:
reduced carrier → electron-transport chain → non-O2 terminal acceptor
                         ↓
                  ion gradient
                         ↓
                    ATP synthase

fermentation:
NADH → organic electron acceptor
          ↓
       NAD+ regenerated
          ↓
glycolytic substrate-level phosphorylation continues

Anaerobic respiration therefore preserves the central respiratory architecture of electron transport plus chemiosmotic coupling. Fermentation does not require an electron-transport chain.

Terminal acceptor choice affects available free energy

The free energy available from electron transfer depends on the redox properties of both donor and acceptor. Oxygen is a powerful oxidizing agent, but it is not the only species capable of accepting electrons in biological systems.

Changing the terminal acceptor changes the overall redox reaction and therefore changes how much free energy can potentially be conserved.

Respiration is broader than mitochondrial respiration

Eukaryotic aerobic respiration commonly uses the inner mitochondrial membrane. Many prokaryotes instead place respiratory electron-transport machinery in their plasma membrane.

The general concept is independent of the organelle:

$$\text{electron donor}\rightarrow\text{ETC}\rightarrow\text{terminal acceptor},$$

with electron-transfer free energy coupled to formation of a membrane electrochemical gradient.

The defining distinction is therefore mechanistic: respiration uses a membrane electron-transport chain and terminal electron acceptor; fermentation regenerates redox carriers without requiring respiratory electron transport.