An intrusive /r/ is any /r/ sound that appears in speech at the end of a syllable where it is not usually found. It is a common variation in many dialects, perhaps being most common in the word 'idear', although in recent years it has also become common in 'Obamar'. This is most common in British dialects, but does occur in American English, particularly in the southern states and in New England.
This is due to an over-generalization of the use of the linking /r/, where dialects that usually delete the final /r/ in words keep it in those cases where it is followed by a word that begins with an vowel. For example, 'bigger' may be typically pronounced /’bɪgə/ -- but when it is followed immediately by a vowel the /r/ is pronounced, as in /’bɪgər ən betə/.
When a speaker is conditioned to avoid word combinations in which a word ending with a vowel is followed by a word beginning with a vowel, it is not uncommon to try to repair this by adding in a consonant. In English this is formally recognized, for example, in the a/an convention. This sort of phenomenon is known as sandhi, although that term includes many phenomenon other than intrusive letters.
Intrusive /r/ occurs most often after the unstressed /ə/ vowel ("the idear of it!") or the vowels /ɑ/ and /ɔ/ ("Drawer another pint"). Recall that pronunciation often does not match spelling; 'Obama' and 'idea' often end with /ə/ in spoken language -- as do many other words that should not if one were to enunciate clearly. Intrusive /r/ may also appear inside words, as in "drawring a picture".
The intrusive /r/ may also appear sometimes when a speaker who is not used to using final /r/ is attempting to over-articulate, or trying to hide their accent.