The R-word is a euphemism for retard and retarded, words that are considered offensive and disrespectful when used to describe or insult individuals with intellectual disabilities or to insult people, places, and things by comparing them to individuals with intellectual disabilities. The campaign against the R-word comes at a time when the term mental retardation is also falling out of use, increasingly replaced by intellectual disabilities or cognitive disabilities in medical and legal language.
Protests over the use of the R-word in the movie Tropic Thunder grew into a movement to remove those words from common speech. Special Olympics started a campaign at r-word.org asking people to pledge to stop using the R-word, with a particular push on March 7 of every year to "Spread the Word to End the Word." The site offers resources for those who wish to encourage others to take the pledge.
Though many folks who defend the use of the R-word claim that such efforts mean "you can't say anything anymore," there are in fact many less-offensive words that work as substitutions. Check my list of 225 options to make your speech more varied and less hurtful.