Unicellular organisms do not breathe in the typical sense, but they respirate by allowing oxygen to enter the cell membrane through the process of diffusion. Because they are so small, they do not have the organs, pores and entryways to breathe like multicellular organisms.
All organisms need to be able to perform some type of respiration to survive. Whether the organism thrives off of oxygen or another type of gas is dependent on the organism, but the process is the same. For energy to be produced in the form of ATP, which is used by all cells, there must be some type of gas exchange. Unicellular organisms are able to transmit their extra gas through the cell wall and take in oxygen and other gasses to help them survive.
There are some animals on the Earth that are bigger than unicellular organisms but still take in their oxygen in the same way. Fish, for example, have gills that allow them to breathe. The oxygen from the water they are "breathing" enters their gills in the same way oxygen enters a unicellular organism. It is then exchanged for the unneeded gasses in the fish's body. Not all animals function this way, but there are some that are able to respirate in the same way that unicellular organisms are without being related in anyway to the organisms.