Everybody staying in the Czech Republic for more than 30 days needs to register! This also includes EU-nationals.
To register, you need to go to the Alien and Border Police. If you spend your time in the Czech Republic in booked accommodation, such as a hostel, hotel etc., the registration will (and has to) be done for you by the person providing the accommodation. Registration is not necessary for people under the age of 15, consular staff or the staff of any governmental organisation recognised by the Czech Republic.
Non-EU nationals with a long-stay visa need to register with the district office of the Foreign Nationals Department of the Czech Police (Cizinecká policie) within three business days of entering the Czech Republic.
If you applied for a Residency Permit you are automatically registered with the Immigration Police when you pick up the permit.