In Islam, there are four conditions of repentance (derived from the Qur'an and Sunnah) which have to be met, as mentioned by the scholars:
1. The person must immediately leave the sin (i.e. they cannot repent from the sin while they are doing it).
2. The person must feel guilt and remorse.
3. The person must make a
firm intention never to commit that sin again.
4. If the sin relates to the rights of another human being, then that right must be returned. So for example, if the sin involved stealing from somebody, then that stolen item must be returned to them.
For the specific sin of Shirk (associating partners with Allah), I heard Sheikh Yasir Qadhi mention that a person should restate their Shahadah (testimony of faith).
You can find out more about repentance in Islam
here.