What happens when one organism is taken out of the food chain? How does it affect others?

It would cause a ripple effect throughout the rest of the web. The impact can vary, but most of the time, it will be noticeable in some way. This is what ecologists call a "cascade".

One example would be the removal of an apex predator, known as a top-down cascade. Predators keep prey populations in check; without them, the prey quickly overpopulate, because they evolved to make lots of babies to compensate for predation. Most prey animals are herbivores, so more of them means fewer plants.

Removing a prey species will also cause problems, but rippling upwards instead of down. Same goes for plants, if there's a species of herbivore that depends heavily on a certain type of plant for food. This is a bottom-up cascade, which is not as common as a top-down cascade, but just as problematic.