Why ask twice, with two different words for doing wrong?
The prayer says chatanu, "we sinned," and then pashanu, "we rebelled." They are not the same word. The Sages in the Gemara (Yoma 36b) define them when they discuss the words of confession. Chet, they say, is a sin done by accident. Pesha is rebellion, and their proof is a verse about the king of Moav rebelling (Melachim II 3:7).
So why are the two words paired with two different titles? Abudarham says it is on purpose. A father looks at a child's deliberate wrongs as if they were accidents, so we say chatanu to our Father. A king looks at even his people's accidents as if they were deliberate, as in "A king establishes the land by justice" (Mishlei, Proverbs, 29:4). So to our King we use the stronger word, pashanu. In his view the title changes how the same act looks.
He also records a reading that others give: "Forgive us, Father, even though we sinned. Pardon us, King, even though we rebelled." On that reading the word "for" (ki) means "even though." Seder HaYom (a guide to daily prayer, first printed in 1599) adds that the prayer asks about the lighter wrongs first and the heavier ones second.
Here is something to notice, and it is ours, not Abudarham's. In Yoma 36b, Moshe asks God: when Israel sin and repent, make their willful sins like accidents. That source does not mention this prayer. But the same idea appears in both: a deliberate wrong looked at as if it were closer to an accident.
Not every teacher charts these words the same way. The Arukh HaShulchan (a law code by Rabbi Yechiel Michel Epstein, around 1900) notes that the Yom Kippur confession pairs the words for forgiveness with different kinds of wrong. Then he points out that in this blessing, selichah goes with chet and mechilah with pesha. So there is no single chart that all the sources share.
Yoma 36b,Abudarham, Weekday Prayers, Amidah,Mishlei 29:4,Seder HaYom, on the Amidah,Arukh HaShulchan, Orach Chaim 607:6Can I just say "forgive us" and be done?
Not if you go by the order of the blessings. Right before this one is the blessing about returning. The Gemara (Megillah 17b) builds the order on a verse, Yeshayahu 55:7: "Let the wicked leave his way ... and return to Hashem, and He will have compassion on him ... for He will abundantly forgive." Returning comes first, then forgiveness.
What does returning mean? The Rambam (Maimonides, the great medieval codifier) spells it out. The person leaves the sin, takes it out of his thoughts, and decides in his heart never to do it again. He regrets the past. And he confesses in words. For the first part, the Rambam quotes that same verse.
The Mishnah (the earliest written record of the Oral Law) closes the obvious loophole. Someone who says, "I will sin and then repent, I will sin and then repent," is not given the chance to repent (Mishnah Yoma 8:9).
Are the words, then, empty? No. The Gemara (Yoma 86b) reads Hoshea 14:3, "Take words with you and return to Hashem," to mean that for a wrong done in private, God is appeased by words. The Ribon HaOlamim prayer says more about that verse. But the verse says "and return," not only "take words."
Seder HaYom adds one more piece for this blessing: when you ask, have the actual wrongs in mind, so that your heart gives way and you pray with more focus.
Megillah 17b,Yeshayahu 55:7,Rambam, Laws of Repentance 2:2,Mishnah Yoma 8:9,Yoma 86b,Seder HaYom, on the AmidahI was cruel to a friend. Does asking God to forgive me fix it?
Not by itself, say the sources. The Mishnah (Yoma 8:9) says Yom Kippur atones for wrongs between a person and God. For wrongs between one person and another, it does not atone until the person makes peace with the one he wronged.
The Hashiveinu prayer gives the Rambam's basic rule: even if you pay back what you took, or only hurt someone with words, you have to go to the person and keep at it until they forgive you.
He also sets a limit. If the person refuses, bring three friends and try again. Do this up to three times. After that you may leave it, and the one who refuses to forgive is the one in the wrong. The exception is your teacher: you keep asking, even a thousand times.
Then he turns it around (Laws of Repentance 2:10). When someone asks you for forgiveness, you may not be cruel. You should forgive with a full heart and not hold a grudge, even if they wronged you badly.
These sources talk about Yom Kippur and repentance in general. They do not name this blessing, so applying them to it is our reading. The Gemara (Yoma 86b) explains why people are harder than God. With a friend, it is uncertain whether he will be appeased at all, and if he is, whether words will be enough. With God, for a wrong done in private, words are accepted.
Mishnah Yoma 8:9,Rambam, Laws of Repentance 2:9 and 2:10,Yoma 86b