Mesosomes are part of some prokaryotic cells and participate in the process of aerobic respiration.
Cellular respiration is the process by which food is broken down and energy is released in the form of ATP molecules. In case of eukaryotic cells, this process takes place in an organelle known as mitochondria (also referred to as the power house of the cell). Prokaryotes do not have mitochondria and hence, have to depend on some other cell component for carrying out this process. Mesosomes are protrusions on the inner side of the cell membrane (as detected by electron microscopy) of prokaryotic cells (such as azobacter, sporulating bacteria, etc.). Mesosomes increase the surface area of the membrane and act as the site for electron transport chain and hence greatly aid the process of aerobic cellular respiration.
Hope this helps.