Electron carrier molecules do just what their name says. They carry electrons from one part of an energy processing system to another, providing the necessary energy and reducing power to make chemical reactions occur. The energy processing systems you are referring to are mainly aerobic cellular respiration and photosynthesis.
In aerobic cellular respiration, the main electron carrier molecules are NADH and FADH2. NADH is produced during glycolysis and the Krebs cycle and then used in the electron transport chain. In the electron transport chain, NADH and FADH2 transfer their electrons to molecules that act as proton pumps. As these proton pumps are reduced by gaining the electrons from NADH and FADH2, they are able to transfer protons across the inner mitochondrial membrane to create a proton gradient that then powers the synthesis of ATP.
In photosynthesis, the main electron carrier molecule is NADPH, which is similar to NADH. NADPH is produced by oxidizing NADP+ during the light dependent reactions. NADPH is then used for reducing power during the Calvin cycle, where it helps power the reactions used to make glucose.