The Arabic root that means “forgive” is غفر (GH-F-R, ghafara). There are a couple of other roots that could be used here but we’re sticking with this one because it’s by far the most common and because it gives us a chance to talk about a couple of grammatical items. The first is the imperative form of the verb, for when you say “forgive me” you’re making a grammatical command. Arabic forms the imperative by taking the second person imperfect, which in this case would be تَغفَرُ (taghfaru), dropping the initial consonant, and dropping that last short vowel. If what remains begins with a consonant followed by a short vowel, then that’s the imperative form. If, however, you’re left with a consonant followed directly by another consonant (as we have here), then a short vowel is added at the beginning of the word. Which short vowel depends on the vowels the verb uses in its normal imperfect form, but let’s not go down that road and just say that, for غفر, the imperative form is اِغفَر (ighfar). غفر takes an indirect object with the preposition ل, so “forgive me” translates to اِغفَر لي (ighfar lī). “I forgive you” would be أغفَرُ لَكُم (aghfaru lakum, or laka or laki if you want to use the singular/informal, gender-specific).
“Forgiveness” has a couple of forms, perhaps غَفر (ghafr), مَغفِرة (maghfirah), or غُفران (ghufrān). The one who does the forgiving is the active participle, غافِر (ghāfir), while the one being forgiven is the passive participle, or مَغفور (maghfūr; “to be forgiven” is the passive voice, غُفِرَ ghufira). Forgiveness being frequently tied to religion, there are several religious uses for غفر. For example, one of the 99 Names of God venerated in Islam is الغُفّار (Al-Ghuffār), The All-Forgiving. You might frequently hear the phrase غَفَرَ الله لَنا وَ لَكُم (ghafara Allāh lanā wa lakum), which means “God forgive us and you!” This sentence uses the optative mood, a grammatical mood used in situations where you wish or hope that something might happen (another example would be حَفَظَ الله المَلِك, ḥafaẓa Allāh al-malik, “God save the King!”). Arabic’s optative is simply the perfect (past) tense. The final item of note here is the Istighfār or Astaghfirullāh, the Islamic prayer of forgiveness, which consists of the Form X of غفر, which is استَغفَرَ, in the first-person imperfect, أستَغفِر (astaghfir). The full phrase أستَغفِر الله (astaghfir Allāh), means “I beg forgiveness of God,” and is uttered over and over in this prayer.