If the father's name ends in -й (like Андрей, Сергей, Тимофей, Николай, etc.), the masculine form of the patronymic ends in an unstressed -евич [jɪvɪtʃ], e.g. Андреевич [ənˈdrejɪvɪtʃ]. In this case in the colloquial language this -евич [jɪvɪtʃ] can be shortened/reduced to -[й]ич [(j)ɪtʃ] with the final -й of the father's name dropping out or not, depending on the speaker, so
Андреевич [ənˈdrejɪvɪtʃ] > Андре[й]ич [ənˈdre(j)ɪtʃ]
Николаевич [nɪkəˈlɑjɪvɪtʃ] > Никола[й]ич [nɪkəˈlɑ(j)ɪtʃ]
If the father's name ends in another consonant (like Степан, Кирилл, Вадим, etc.), then the masculine form of the patronymic ends in an unstressed -ович [əvɪtʃ], e.g. Степанович [stɪ'pɑnəvɪtʃ]. In this case in the colloquial language this -ович [əvɪtʃ] can be shortened/reduced to -ыч [ɨtʃ], so
Степанович [stɪ'pɑnəvɪtʃ] > Степаныч [stɪ'pɑnɨtʃ]
Кириллович [kɪ'riləvɪtʃ] > Кирилыч [kɪ'rilɨtʃ]
This, however, does not apply to some names, e.g. although Олег ends in a consonant, you can hardly hear somebogy saying *Олегыч, since the sound combination гы is normally not found in Russian.
Also, Петрович has no reduced form, since Пётр is a monosyllabic name and the suffix -ович is stressed in it, so it cannot be reduced.
And some names have irregular colloquial patronymics mentioned by Schullz and user4419802 in the comments to the question:
Алексáндрович > Сáныч
Михáйлович > Михáлыч
Пáвлович > Пáлыч