I have always thought that Aleinu was a piyyut (liturgical poem), not a "prayer" since it does not have a bracha in it. I was surprised to find that the Wikipedia page for piyyut does not list it, and that the page for Aleinu calls it a prayer. I realize that Wikipedia may not be the most authoritative source for Jewish answers, so I am asking it here. Is Aleinu considered a piyyut or a prayer? Either way, why? A related (unanswered) question is: Why are Ashre and Alenu considered communal prayers?
Actually, a text can be both a prayer and a piyyut. A prayer is a type of content, whereas a piyyut is a type of style. A prayer is when one talks to God (OED, MW); it need not contain a bracha. A piyyut is a style of a text (which may or not be a prayer) similar to poetry.
Therefore, there are selections which are prayers, piyyutim, both, or none. Mi Sheberach is a prayer (note the absence of a bracha), but not a piyyut. Yigdal is a piyyut, but not a prayer. Veye'esayu (on Rosh Hashanah and Yom Kippur) is a piyyut and a prayer.
Specifically, the second part of Aleinu is a prayer (since we acknowledge God as the ruler of all, and pray that He be universally recognized as such soon). It can be argued that since part of it is a prayer, the whole thing is. It is not a piyyut.