Aleinu has a fraught textual history. The original text contains a phrase praising God "who did not make us like the nations... who bow to vanity and emptiness and pray to a god who does not save". In medieval Christian Europe this phrase was read as an attack on Christianity. From the 13th century onward, Christian censors repeatedly demanded its removal; many Ashkenazi siddurim from the 15th-19th centuries had the controversial line struck through or omitted entirely.
The phrase is biblical, it's a direct quote from Isaiah 30:7 and 45:20, referring to idol-worshipers in general (not Christianity specifically). Modern Israeli Ashkenazi siddurim have restored the original text; some diaspora siddurim still omit it. Sephardic, Mizrahi, and Yemenite siddurim never omitted it.
The second paragraph (not on this page) is the more universalist one. After establishing Israel's particular relationship to God in paragraph 1, paragraph 2 looks forward to a future in which all humanity will recognize God as one. Key line: "l'takken olam b'malchut Shaddai", "to perfect the world under the kingship of the Almighty". The phrase tikkun olam ("perfecting / repairing the world") that has become a Jewish slogan in the modern era comes from this paragraph.
The paragraph ends with a quote from Zechariah 14:9: "And the LORD will be King over all the earth, on that day the LORD will be one and His name will be one." The same word echad (one) that the Shema affirms about God's nature today is echoed here as a future-tense political-cosmic claim about the recognition of that oneness.
The closing sequence of every daily service: Aleinu → Mourner's Kaddish (if mourners are present) → Adon Olam or Yigdal (closing hymn). Aleinu is the universal core; the others depend on context. If you walk into a synagogue and people are standing for Aleinu, you know the service is wrapping up, about 5 more minutes.