I know this question has been answered already, but at least one didn't nail your question I don't feel. Rhotacism is specifically pronouncing an R in circumstances where you wouldn't otherwise. In other words it's the pronunciation of the R when certain letter and phonemic combinations are found. Also, it's not the standard English R pronunciation, it is the alveolar tap or trill, found in languages like Italian, Spanish, etc. (Also found in some English, but not standard). These changes in English where we might make an alveolar tap or trill appears in informal things like:
"I got to get it."
"I goda gedit"
^ ^ // Pointers point to alveolar taps or rarely trills
Rhotacism is also the impairment of not being able to make this sound, notice that the Wikipedia article says that the people who have this condition are usually the:
Finnish, Hungarian, Italian, Indonesian, Bulgarian, Romanian, Russian
Which shows that it's not this sound that English speakers make when they pronounce Rs in words.
The real phenomenon, of NOT pronouncing the R at least in English is:
Rhoticity in English
non-rhotic was given in the other answer, but what the heck I've added it here anyway, plus it's also the noun form, which is what you asked for.