There is no short answer to this question, here is a full account of what is known within the Star Trek universe about the Illyrian-born human Starfleet officer:
The name of the Star Trek character seen in Star Trek:Discovery as well as Star Trek: The Original Series has long been controversial however in Star Trek: Short Takes episode Q&A her official name is displayed on her PADD confirming that her official Starfleet name is Una during the time when she served as Captain Pike's First Officer.
In this same episode however she told Spock that
My name is Number One.
These statements are not necessarily contradictory:
Una is a girl's name in the English language derived from the Gaelic word for "unique." In many languages the word itself means "one":
Una is the Latin feminine word for numeral "1" (literally, it means "the number 1.")
Una is the Spanish feminine singular of the adjective uno = "One"
Une is the French feminine singular of the adjective un, and it is pronounced OO-na"
Una means "One" in Irish.
Long-time Star Trek writer Dorothy Catherine "D. C." Fontana explained the origins of this name in her novel Vulcan's Glory, which states that this character's moniker was not simply a nickname or title – she was a product of Illyrian genetic engineering who was called "Number One," as the best product of her batch.
As such, the name "Number One" is a literal English translation of "Una" so both statements are technically correct.
LCDR Una's rank and title:
At various times onboard Enterprise she is adressed as simply "Lieutenant" rather than "Number One" or "Lieutenant Commander." This is common the in Naval forces which Star Trek rank structure emulates. In the same way that "Captain" is both a billet title, which simply means it is your job to "captain" the vessel, or the same word is also a military rank; Lieutenant is both a position title and a rank. The military billet title 1st Lieutenant (abr. 1Lt) literally means that this person is the "First Officer" in succession of command, also known as the Executive Officer (XO), or "Number One," and occasionally just "Lieutenant." When the Captain addresses her as "Lieutenant" he is not using her rank, he is referring to her job. Her military rank aboard the Enterprise has always been Lieutenant Commander (LCDR) and as the script was written her job has always been to be the Executive Officer (The Making of Star Trek, pp. 22-30)
LCDR Una's given name on Illyria is impossible to pronounce according to some novels and it has not yet been revealed within a canonical source.