Mitochondria are an organelle of many eukaryotic cells, such as human cells, for example. They can be thought of as the power plants of the cell because they produce most of the Adenosine TriPhosphate (ATP) used by the cell which is a key source of chemical energy for the cell's activities. But mitochondria are also implicated in other functions like controlling cell cycle and growth, differentiation, and death as well as signalling functions. Although having largely positive effects mitochondria have been implicated as factors contributing to mental illnesses and aging.
Mitochondria, on one theory, (known as endosymbiotic theory), are thought to be the remnants of an ancient bacterial species that invaded or were swallowed by the forerunners of present day eukaryotic cells. But instead of one species destroying the other as you might expect, they found instead that they could provide each other with some mutual benefit, becoming symbiotic species. This theory is backed up by the fact that mitochondrial DNA is passed down independently from that of the nuclear DNA of the cell.
Structurally mitochondria can be seen as a cylindrical organelle, around a quarter of a micrometer long, with a matrix surrounded by a double membrane. It is compartmentalised and each compartment performs a specialized function. The outer mitochondrial membrane surrounds the whole mitochondrion and limits what can enter and leave, for example.
The intermembrane space, between the outer and inner membranes, see localised build-ups of proteins such as cytochrome c, for example. The inner membrane has proteins that can do several things, such as oxidative phosphorylation, ATP synthase, metabolite regulation, and protein importing, for example. The fourth compartment, the cristae, improve ATP production by providing increased surface area on the inner membrane. The final compartment, the matrix, is the space inside the inner membrane with two thirds of the proteins of the mitochondrion in it. The matrix contains mitochondrial ribosomes, transfer RNA, and multiple copies of the mitochondrial DNA itself.
The presence of mitochondria in cells varies very much both across different organisms and across different types of tissue within the same organism. This difference in numbers can even range across around four orders of magnitude with some cells having only one mitochondrion whilst other cells may have in the thousands. The liver, for example, is known to contain cells with thousands of mitochondria inside them. At this level they make up around 20% of the cell's volume.