Pasture is land with grass growing on it for farm animals to eat.

If someone leaves for greener pastures, or in British English pastures new, they leave their job, their home, or the situation they are in for something they think will be much better.

If you put animals out to pasture, you move them out into the fields so they can eat the grass.

If you say that someone is being put out to pasture, you mean they are no longer being employed because they are considered to be too old or no longer useful.