I know that idea is pronounced as /aɪˈdiə/, but I've meet several people in real life who put an 'r' at the end of the word. How come?
In practically all dialects of (British) English, the word "idea" would generally be followed by an 'r' sound when followed by another word beginning with a vowel. So for example in saying "it was his idea and decision", this would usually be pronounced "idea-r-and decision". The phenomenon occurs in many places where you have a word ending in a vowel followed by another word beginning with a vowel and they are 'closely linked' syntactically, e.g. "law-r-and order", "Canada-r-and America", "he took the law-r-into his own hands" etc.
This occurs essentially because it's a natural phenomenon of language to have some strategy for 'linking' words together. These strategies tend to make it more likely for syllables to have a consonant in their onset, and there's some evidence that languages universally "prefer" this. French has a similar phenomenon involving 'latent' consonants (principally [z]), Spanish allows vowels to 'merge' into the same syllable, etc.
Update: there seems to be some disagreement about the exact extent of this phenomenon. In his article in the Blackwell Handbook of English Linguistics, Michael MacMahon (2008) describes it as:
"now very common in many accents of non-rhotic British English"
though without giving any actual data. Gick (1999), one of the articles referenced by the Wikipedia entry mentioned in Peter Shor's comment below, observes:
"[...] the available data, which has centred almost exclusively on only two dialects: Southern British Received Pronunciation [...] and the dialect of eastern Massachusetts [..]. No other r-intruding dialects have been described which provide evidence bearing conclusively on the controversy."
However, I think by "the controversy" here, the author means "how the phenomenon is to be analysed phonologically" rather than whereabouts the phenomenon actually occurs. What seems to have happened is a few studies have focussed on certain specific dialects, but we shouldn't conclude from that that the phenomenon occurs only in those dialects.
Adding r's to the end of words is something odd I first noticed as a child with my grandmother. Idea became "idear," "Ella" became "Eller," etc. I assumed it to be part of her Southern upbringing until I heard Billy Joel sing about "Brender and Eddie" in the "Ballad of Brenda and Eddie" segment of his "Scenes from an Italian Restaurant." Apparently this goes up into Maine!
This most likely has its roots in hypercorrection. Much as some Cockneys began to worry about "dropping their h's" such that some adding them to the beginnings of words that actually do start with vowels, one can observe that R's are dropped at the ends of most words in Received Pronunciation British English. Accents from the Atlantic states, whether Northern or Southern, ultimately derive from England, and so this probably originated with people from less literate times adding r's in their heads to words that didn't actually end with them. After all, they were used to not hearing the final consonant in plenty of words that actually DO end in r's.