In the United States at least it is very offensive to refer to people with physical or mental abilities with this word. This is a word bullies use.

I will assume you are referring to its use regarding a person. Family members of someone who is mentally retarded often do not want their loved one referred to as retarded because of the stigma associated with the word. I read a lot of medical charts and the term is used by doctors; they often opt to abbreviate "MR" instead of writing mentally retarded, but they abbreviate almost everything. I think in other places like the media they say mentally challenged which I guess is the politically correct way.

It depends on the context. It is offensive and pejorative when used as a noun to describe a person who has some physical or intellectual disability or to suggest that a person has such characteristics. When used as a verb, it simply means to slow down or hinder something. "The brakes on the vehicle retard its speed". "His decision has retarded the progress of the project"

Because to say somebody is mentally ill when they are not is considered rather rude. It implies they are abnormally, clinically unintelligent or that their behaviour is in some other way abnormal.