What's the right term/name for a person who is an expert eg. a smith, a plumber, electrician, a writer, clerk etc but he or she didn't went to college/university to study that particular course. And how should they describe themselves when writing a business/application letter?
I think the word you're looking for is self-taught
having knowledge or skills acquired by one's own efforts without formal instruction
a self-taught musician
One wouldn't usually use this term in a title, but you might mention it in a resume/CV, to indicate your initiative.
For most trade professions "master" is appropriate (as @AndrewLeach stated) like "master plumber", "master electrician", "master brick layer", "master jeweler", etc.
For other jobs the word "professional" works. Some examples are: "professional writer", "professional athlete", "professional clerk", etc. This doesn't alway connote a high level of expertise but for the purpose of using it as a signature it fits.
For some jobs that have organization which administer tests to become a professional the word "certified" or "licensed" is appropriate. Examples of that are: "certified public accountant", "certified management accountant", "certified public bookkeeper", "licensed real estate agent", "licensed professional counselor", "licensed attorney", etc.
This answer is not exhaustive but covers most of the more common titles for jobs.
The answer is in the title. Not the question title, but the trade title.
When discussing a trade, a person cannot claim the title until they are at least somewhat experienced at that trade. Someone who is in process of learning a trade is an apprentice, undergoing an apprenticeship, or else they are teaching themselves.
There's been suggestions that speak about mastery, but the question of expertise in a trade has more to do with whether a person has done certain things with the knowledge of the trade. I couldn't call an expert carpenter a master carpenter unless they have had an apprentice, as there is a relationship between master and apprentice in that you require an apprentice to be a master, and you require learning from a master to be an apprentice.
So, really, you are either, in order from least important to most:
- Not a tradesman
- An apprentice
- A (novice, experienced, or expert) tradesman (plumber, carpenter, mason, etc.)
- A master of a trade