Evidence for the endosymbiont theory is strong. Most convincingly, mitochondria closely resemble extant (currently existing) bacteria. Mitochondria are similar in size, and they replicate in a fashion that is very similar to binary fission. Also, the inner membrane of mitochondria bears a strong resemblance to the membranes of prokaryotes, sharing several key proteins and transport systems. Additionally, mitochondria have their own DNA (which takes the form of a circular plasmid, like that of prokaryotes) and they possess all of the cellular machinery required to transcribe and translate their genomes, thereby enabling them to produce their own proteins.
The character of mitochondrial DNA is much like that seen in modern-day prokaryotes. For example, mitochondrial DNA lack histone proteins, which are associated with eukaryotic nuclear DNA but not with prokaryotic DNA. Additionally, mitochondrial ribosomes are more similar in behavior, structure, and nucleic acid base sequence to the ribosomes of prokaryotes than they are to eukaryotic ribosomes; there is high sequence similarity between the ribosomal RNA of mitochondria and that of modern endosymbiotic bacteria.
The relationship between mitochondria and eukaryotes has grown so intimate that neither can exist without the other (in other words, this has become an "obligate" symbiosis). Most mitochondrial genes are contained within the nucleus of their eukaryotic host cells, which makes mitochondria unable to reproduce and survive independently. Likewise, eukaryotes (e.g., humans) would be unable to manufacture enough ATP to sustain life without the help of mitochondria.
Sometime before 2.2-2.3 billion years ago, cyanobacteria evolved the ability to use H 2 O and CO 2 to make organic molecules with the help of solar energy, in a process known as photosynthesis. The primary "waste" product from this type of photosynthesis is O 2 , which began to accumulate in the atmosphere due to the activity of cyanobacteria. Other earlier forms of photosynthesis probably relied on H 2 S rather than H 2 O and did not result in the release of O 2 . For millions of years, cyanobacterial photosynthesis did not change the Earth's atmosphere; oxygen released by photosynthesizing mats of marine cyanobacteria combined with iron ions in the ocean, forming iron oxide that precipitated to the sea floor. When these iron ions were depleted, oxygen began to accumulate in seawater and eventually it diffused into the atmosphere.
The change in atmospheric composition due to cyanobacteria was enormous, even more severe than the pollution associated with industrialized civilization. Oxygen is a powerful oxidizer, and its tendency to strip electrons and attack the bonds of organic molecules can be very dangerous to living organisms. The atmospheric changes effected by cyanobacteria probably resulted in numerous extinctions, but conversely, they also led to novel adaptations (e.g., the production of antioxidants). Furthermore, some organisms also evolved the ability to detoxify oxygen by reducing it with electrons from other molecules. Electron transport chains, such as the ones that exist in modern-day mitochondria, might have evolved as mechanisms for counteracting the destructive effects of O 2 , secondarily becoming a means of energy production over evolutionary time. The ancestors of eukaryotes might have gained an advantage in an oxygen-rich atmosphere by adopting endosymbionts to detoxify O 2 . The additional benefit that eukaryotes enjoy today, ATP synthesis, might be a derived function.
This video (from Neural Academy) nicely explains the theory of endosymbiosis as well as the evidence for endosymbiosis:
To watch this video on YouTube (and see closed captioning) - press the arrow icon in the bottom right corner of the video player.