As an EU National, you do not need a visa to enter the Czech Republic, and you can stay in the country for up to 30 days without having to register with the Foreign Police.
If your intention is to stay in the Czech Republic for longer than one month, then you are obliged to register with the Foreign Police within your first 30 days here.
As an EU citizen, you do not have to apply for any kind of residence permit. However, there are certain life situations when you might be required by the authorities to apply for Temporary Residency in the Czech Republic. Such situations may include: