You have found a special form of articles in German, called the "Nullartikel". This translates roughly to "null article" or "zero article" and can be seen as a word with a length of zero letters - so in effect, the article is left out. Though you probably could debate for hours whether a word with zero letters actually does exist or not ;)
There are some cases where the Nullartikel can be used, for example
Names, salutations and academic titles of people
Das musst Du Stefan fragen.
You have to ask Stefan about that.
Die Abteilung wird von Frau Müller geleitet.
The department is headed by Ms Müller.
Das neue Buch von Professor Schmidt ist sehr interessant.
The book by Professor Schmidt is very interesting.
Professions without further qualification
Er arbeitet als Kellner.
He works as a waiter.
but
Er ist ein erfahrener Taxifahrer.
He is an experienced cab driver.
The plural form of the indeterminate article
Ich lese ein Buch. -> Ich lese Bücher.
I read a book. -> I read books.
Names of countries, citys, continents, nationalities, ...
Ich reise nach Deutschland.
I travel to Germany.
Though there are some exeptions here:
Ich reise in die Schweiz.
I travel to Switzerland.
The use cases above are just examples, there are others. As you've probably seen in the examples, there are cases where the English language doesn't use an article either, while in other cases the handling differs between German and English. You can get quite into the weeds with the Nullartikel, especially when it comes to the exceptions of the rules.
Regarding your specific example,
Er ist Kellner.
and
Er ist ein Kellner.
are both grammatically correct and have just about the same meaning. There's a small difference in nuance: "Er ist Kellner" reads more like being a waiter is his profession, his calling in life maybe, while "Er ist ein Kellner" has more of a vibe of a job that maybe changed next month to something else.
