Shinigami or Soul reapers are protectors of the real world, they exist to combat "hollows"

an example of shinigami below
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an example of hollow below
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Hollows come to the real world through rips they find in space, they come completely by accident, but seize the opportunity to fiest on the souls of either the living, or those at unrest, meaning the spirits of those who have unfinished business and linger as ghosts

a shinigamis job is to protect this ghost by slaying the hollow with Zanpaktou, or swords that have hidden powers, each zanpaktou can be "released" by learning the name of the swords inner spirit, a released zanpaktou has various powers, and releasing it normally changes the shape of the sword (a zanpaktou is just a regular katana)

below is a picture of what the katana looks like for ichigo after he releases it
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after a shinigami slays a hollow it is his or her job to then send the lost soul to the soul society, or serietei, the place where shinigami live and operate in, soul society has 13 squads of shinigami that each have a unique function, they work together to keep balance in the world

a shinigami also has access to "kido", these are powers used without relying on a sword, they can vary from firing lightning to firing fire, but they must have an incantation behind them, only skilled shinigami can do it without incantation

kido relies on Reiatsu "spiritual pressure" this is basically the shinigamis power source for all attacks including release and bankai, spiritual pressure actually makes the world feel heavier to those around you, but only mentally

lastly, shinigami have after a release, "Bankai" which is a second release of the sword, giving the user immesurable strength and power, however few ever attain bankai, it is said to take a thousand years however ichigo from the anime managed it within a few months

for all this and more, the first episode of bleach explains alot ^^