Persians are the people of Persia which today is called Iran. It is also common to call these people Iranians, Irani, Aryans, or Aryani.
I have Persian descent through my mother.
Persians on Persians
It is important to note that Persians don't actually call themselves "Persians" and don't call their country "Persia".
Persians consider "Persia" (in this context) to be the old name of their Political State. And a "Persian" to be a citizen of that ancient Political State.
Persians call themselves Irani; which in English becomes Iranians. The older pronunciation of Irani is Aryani; which in English becomes Aryans.
However, in English, it is correct to call these people Persians. Just as it is correct to call Germans "Germans", even though Germans don't call themselves that, but call themselves "Deutsch".
The Persian Language is called Farsi or Parsi by Persians. In English it is appropriate to just call it Persian.
The Iranian province of Fars was once called Pars. At one time, this province was politically dominant. Because of its political dominance, the Greeks called Iran and all the territories controlled by them after the province of "Pars". Which English became "Persia".
Which is why today, Iran is called Persia in English. And why the Iranians are called Persians.
When people think of Persians, many think of the ancient Persian civilizations. Some also see Persians as creators of modern Medicine or modern Mathematics. And other think of the Poetry and Literature from Persians.
Persians have a reputation for 3 things. #1: Persians seem to breed Mathematicians. #2: (Male) Persians tend to be very muscular. (There's a high percentage of Mesomorphs among Persians as compared to other populations.) #3: Persians have a reputation for being fierce warriors.