Diffusion is the method in which amoeba obtain oxygen and expel waste gas (carbon dioxide). That is, diffusion is used as the method of respiration in amoeba.
Because the surface area of an amoeba is large compared to its volume, diffusion is an efficient means of oxygen intake and expelling carbon dioxide through the plasma membrane. This same process also occurs through the (moist) skin of simple creatures, like earthworms. For larger creatures, the surface area is not large enough for diffusion to be effective: specialised organs (gills, lungs) must be employed to service the whole organism.
Diffusion requires no energy input: molecules move from an area of high density to an area of low density, such that an overall equilibrium concentration is reached.
Amoeba will also take in water through osmosis, however, unlike the diffusion of gasses, excess water needs to be forcefully (mechanically) removed.