I'm drafting a letter of invitation for someone. And one sentence goes: "I am [the / an] owner of a bookstore" - which article shall I use?
If the bookstore you own has only one owner (you), then, "I am the owner of a bookstore." is correct.
If there are other owners of that bookstore (i.e. you are a co-owner), then you should say, "I am an owner of a bookstore."
The indefinite article, "a" is for one among other(s), and the definite article "the" is for naming one when there are no others.
It depends on context.
If you are talking about a particular bookstore then it depends on whether you are the sole owner or a part owner. If you are the only owner of that bookstore:
I am the owner of a bookstore
if you and other people jointly own it:
I am an owner of a bookstore
These fragments would likely then contain more specific information about that particular bookstore, for example: I am the owner of a bookstore located in Main Street ...
If you are talking about yourself:
I am an owner of a bookstore
This is correct if you are the sole owner. If you and other people jointly own it, you would say something like: I am a part-owner of a bookstore. In this case the remainder of the sentence would likely contain information about you, for example: I am an owner of a bookstore who thinks small businesses are unfairly taxed.
If you introduce yourself by mentioning your occupation (what I am):
I am an engineer.
A step towards your example:
I am a bookstore owner.
The article 'a' above defines the noun 'owner'. Next step (considering it's still about your occupation, not a particular store):
I am an owner of a bookstore.
Like most things, it depends on context.
As most of the other answers have pointed out, if you are the only owner of the bookstore, you'd normally say "I'm the owner of a bookstore." If you jointly owned it, you'd normally say "I'm an owner of a bookstore." Having said that, most people you're talking to probably don't care whether you're a sole owner or joint owner, so "I'm the owner of a bookstore" is mostly going to be OK even if there are other owners. Most of the time, all you need to do is linking the concepts of "I", "bookstore" and "owner".
But there are also cases where you might use "an owner", even if you're the only one. Suppose, for example, that you wanted to go to a trade fair for bookstore owners. At the door, the security guard says to you, "I'm sorry only bookstore owners can come in." You might respond "But I am an owner of a bookstore!" The point here is that you're saying you're a member of the category "bookstore owners" and you're not the only member of that category, so "a" is more appropriate. Even in this case, "But I am the owner of a bookstore!" would be fine – you're focusing on your bookstore, rather than bookstores in general, and the meaning is still "I own a bookstore, so let me in."