Before reading this post, you should be familiar with the International Phonetic Alphabet and Phonemic Alphabets. You may also want to read IPA: History, Functionality, and Limitations as well.
The shower answer to this question is that there is only one International Phonetic Alphabet, the one created in 1888 by the International Phonetic Association. However it is easy to get confused, as for every accent and dialect, there is a different phonemic alphabet. A phonemic alphabet is a simplified form of the International phonemic alphabet, so uses a subset of the total IPA sounds that correspond to the specific sounds for that dialect.
IPA as a whole currently uses 107 symbols for vowel and consonant sounds modified by 31 diacritics (the marks around letters to change the sound, such as å). Additionally, there are 17 more signs that show things like stress, intonation, and length of pronunciation. While there have been some revisions since 1888 (notably between 1890s to 1940s, 1989, and minor ones in 1993 and 2005), these have primarily been additions rather than changes as linguists identify more sounds.
As you can see, there are a lot of numbers; 1 IPA, innumerable phonemic alphabets, 107 basic pronunciation symbols, 31 diacritics, and 17 additional modifying symbols. It is no wonder you were confused about the question “how many”.
Another confusing aspect is that there is not just one chart that shows all sounds. This is usually because not all sounds are needed, so charts that correspond to phonemic alphabets are more numerous. Additionally “IPA charts” are often adapted to the context they are needed, so aspects are left out or moved around depending on contextual usefulness or theoretical relevance. Here is an example from the IPA Wikipedia page:
Image note from Wikipedia: The authors of textbooks or similar publications often create revised versions of the IPA chart to express their own preferences or needs. The image displays one such version. All pulmonic consonants are moved to the consonant chart. Only the black symbols are on the official IPA chart; additional symbols are in grey. The grey fricatives are part of the extIPA, and the grey retroflex letters are mentioned or implicit in the Handbook. The grey click is a retired IPA letter that is still in use.