If you are describing something (i.e., a noun) that is “first in position, rank, or importance,” then the adjective premier is the word you want.
The area’s premier golf club offers members many amenities.
If you are discussing a “first performance or exhibition” (e.g., of a play or motion picture) then the noun premiere is the correct choice.
The show’s Broadway premiere was last night.
In recent decades, the use of premiere as a verb has gained acceptance when used within the context of the entertainment industry.
The star’s new program will premiere this fall.
(Note that the New York Times has not adopted this usage. See The New York Times Manual of Style and Usage, 5th ed.)
The noun premier refers to a government official in some countries.
According to Merriam-Webster’s online dictionary, premier also means “first in time” (i.e., earliest). Hmm, that’s beginning to sound a bit like a “first showing,” and, indeed, the dictionary says that premiere is an alteration of premier. Further, the “chief actress of a theatrical cast” is a premiere. Doesn’t that seem very similar to “first in position, rank, or importance,” the definition of premier, without the -e? We have to look at the origins of both words to understand the confusion.
Premier was the earlier (French) word (meaning “first”), derived (in the15th century) from the Latin primarius (“of the first rank, chief, principal, excellent”).
The feminine form of premier, première (in French), also meaning “first,” appeared in the 19th century.
English, of course, has developed its own rules—which change over time.
So what are we to make of all that?
Don’t use premiere to refer to a government official.
Don’t use premier as a verb. (Although, when writing in the past tense, the result will be the same):
The show premiered on Broadway last night.
Using premiere (as a noun) to refer to a first showing is clearly correct, as is using premier to refer to something (or someone) that is “first and foremost.”
Just as the verb premiere has become widely used by the entertainment industry, premiere is sometimes used as an adjective by marketers who are advertising something. (“We are a premiere medical spa in Anytown, USA”). That usage may gain acceptance in time. (A recent article in Fortune magazine noted: “Bordeaux, of course, is one of the premiere wine regions in France,” but I frequently find spelling errors in publications.) For the present, premier is the correct spelling of the adjective.