Examples of political correctness:
Retarded: Mentally challenged. Even this term is under "PC" review, and was not too recently changed from "handicapped", which was found to be too un-PC.
Black: African-American.
Indian: Native-American.
Moonbat: Liberal. JUST KIDDING.
Essentially, PC language encourages a level and fair playing field, with person-first language, which focuses more on a person than their disability, skin color, et. al. There is to be no sense of degradation, and in PC language, this is expressed to a fault sometimes. Big example is the word "crippled." That term is offensive to some, in the context of person-first language which identifies one as a human being before it identifies them as their ailment, and so "handicapped" was ushered in by Senator Bob Dole. Since then, "handicapped" was found to be offensive, and some folks who are trending in the PC realm refer to the handicapped as "differently abled" and/or "physically challenged." Earlier in history, "negro" or "colored" were seemingly acceptable terms, but now that these have been deemed to be non-PC phrases, "African American" or "person of color" has been ushered in.
The problem a lot of people have with PC language is the issue of free speech. None of the above may seem particularly stifling, but consider the debacle on "Freedom Fries" we had not too long ago, or how some teachers in public schools are telling children not to call them "Easter Eggs" but "Spring Spheres." Yup. Spring Spheres. That Christian language is too non-inclusive.