Well, one night some weirdly thumbsize-like and bacteria-looking parasites are falling from the sky. They infiltrate a human's brain through the ears or nose and devour their brain. What we then have is an emotionless human being who is practically dead and being controlled by the parasyte from the brain. Depending on the parasyte's intelligence and sense of cooperation, Parasyte's try to disguise and adapt their hosts' lives. They can eat humans, but it's not a must. One parasyte tries to invade the main character's body Shinichi Izumi, but fails and ends up entering his right arm instead. From now on, Shinichi and Migi, the parasyte's name, are trying to coexist and keep their special case hidden. As time goes on, Shinichi and Migi sort of learn from each other's race and adapt some of the other parties' personalities. This anime is incredibly deep and adresses some very mature themes (you'd be surprised if I tell you the embarrassing things happening in this anime). Of course, fighting scenes and action are a part of this anime, but it mainly revolves around Migi.
What I want to know is the difference between that and manga.