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The mitochondria are the organelles that are responsible for most of the production of adenosine triphosphate (ATP), the substance that powers nearly all metabolic reactions. Mitochondria are composed of two layers of membrane:

The outer mitochondrial membrane is very permeable (contains transmembrane proteins called Porin which allows many compounds to pass through, like NADH and ATP and pyruvate).
- Note that porins are a general class of transmembrane proteins that facilitate diffusion across membranes.
But the inner mitochondrial membrane, which forms the cristae, is very selectively permeable. Substances like NADH and ATP and pyruvate need to be transported across this second membrane via chemiosmosis.

More specifically, the mitochondrion is responsible for the second and third stages of aerobic (oxygen-requiring) respiration, the first stage being the responsibility of the enzymes in the cytoplasm that break down glucose molecules (glycolysis). At best, 36 -38 ATP molecules can be produced from each glucose molecule.

Mitochondria also function in heat production (the by-product of oxidative phosphorylation). They also store calcium ions (secondary to the Smooth ER) and are believed to be responsible for cellular apoptosis.

Mitochondria can be found in all eukaryotic cells. Mitochondria (and plastids) are believed to have originated in eukaryotic cells by way of endosymbiosis (with mitochondria, its mutualistic aerobically-respiring bacteria phagocytosed by a eukaryote). Here’s some supporting evidence:

Mitochondria and plastids have two layers of membrane.
Their inner membranes have enzymes and transport systems that are structurally very similar to those of prokaryotes.
Mitochondria and plastids have their own DNA: a single, circular chromosome that is not associated with histones or other proteins.
They have their own tRNA and ribosomes to translate their DNA into proteins.
Their ribosomes are more like prokaryotic ribosomes than eukaryotic ones in terms of size, nucleotide sequence, and sensitivity to some antibiotics.
They divide by themselves via fission, independent of the rest of the cell, rather like prokaryotes.