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.
IPA may be great for languages, but who actually uses it? The short answer is that the following types of people/professions may use IPA to varying levels in their pursuits:
Linguists and IPA
Linguists, or a person who studies languages and their structures, use the International Phonetic Alphabet (or a simplified form called a Phonemic Alphabet) when transcribing for a variety of reasons. Examples include when decoding a new (new = new to linguistics) language, comparing pronunciation across dialects, and seeing how pronunciation has changed or varied between generations (e.g. Great Vowel Shift).
Language teachers and Foreign-language learners
Some language teachers will use it in their classrooms, so some (but not all) foreign-language students may see it. In English language classrooms, the schwa (ə) is most commonly taught as it is an important and common sound.

Dictionaries + IPA
Lexicographers are probably the people who are most likely to use the International Phonetic Alphabet regularly as using it has become standard in most well-known dictionaries. Examples of dictionaries in English that include IPA are the Cambridge Dictionary and the Oxford Learner’s Dictionary.
However, many American dictionaries do not include IPA transcriptions. The argument is that most English speakers do not know IPA, so they use other methods to show pronunciation. These include respelling words (e.g. re-SPELL-ing) and dïácrîtìc mårks.
Speech-language pathologists
While not usually used directly with the person they are training, speech-language pathologists (SLPs) usually use the International Phonetic Alphabet when communicating with others and for materials. They use these sounds as target sounds for pronunciation training.
Here are some common sounds used by SLPs (from HandyHandouts):

Actors and singers
When learning lines or a song in a language that an actor or singer does not know, the International Phonetic Alphabet can be a useful way to learn the proper or desired pronunciation. The actor/singer may use it directly or work with a speech trainer/vocal coach (who uses IPA) to achieve the desired pronunciation.
Constructed language creators
Constructed language creators, or people who make up languages (e.g. for fun or media such as Star Trek’s Klingon Game of Thrones’ Dothraki), may also use IPA to create the pronunciation system. This is especially true of languages that go on to be popular, such as the two listed previously. The Wikipedia articles for both have pronunciation tables that include IPA at the time of writing.

