Is -in' really correct? [duplicate]

The 'g' in -ing is never pronounced. What is pronounced is the velar nasal consonant represented in IPA as [ŋ]. In some dialects, this is replaced by the alveolar nasal consonant represented in IPA as [n]. This is the phonetics that the -in' ending represents.

The difference between [ŋg] and just [ŋ] can be heard in the difference between the words finger and singer.

You should never use a [g] in meeting. Use [ŋ] (which is usually represented in English spelling as 'ng') and not [n], [ŋg], or [ng].

The local dialect in several regions of the U.S., and apparently in parts of Britain as well, uses [ɪn] rather than [ɪŋ] for the suffix -ing. This is sometimes spelled -in'. The people speaking these dialects can pronounce the consonant [ŋ] just fine; for example, singin' would be pronounced [sɪŋɪn]. For more information on this, see this dialect blog posting.

Using an apostrophe in place of a "g" is an informal colloquiallism. It is usually found inside a quotation, suggesting the speaker did not use much care when enunciating. Specifically, it's used to indicate the verb was spoken such that the final "g" was omitted: "We were walkin' to the store, not botherin' nobody, when, all of a sudden, out o' nowhere, this guy starts a-hollerin' at us for no good reason!"

(It's similar to using o' in place of "of " - as found in the preceding example).

Sometimes it's also used in song titles, when the singer doesn't carefully enunciate the final "g" in an "-ing" verb (a la "Takin' Care of Business" and "The Times They Are a-Changin'").