More forms of residence (temporary, permanent, tolerated) are applicable for foreigners living in Slovakia. Key powers in this field have Department of the Aliens Police/Foreign Police Department (Slov. Cudzinecká Polícia), as a part of state police forces. Also European Union citizens staying in Slovakia for more than three months are obliged to apply for registration of residence. Usual purpose for temporary residence is study, business or work. Foreigners must be aware that persons applying for residence or its prolongation are paying fees for this procedure. 
For more details you can see:

Address of Bratislava branch of Foreign Police Department serving all foreigners residing in Bratislava:
Oddelenie cudzineckej polície PZ Bratislava 
Regrútska 4 (City quarter Vajnory)
Bratislava 831 07
GPS: N 48.2098064 E 17.190732000000025
Phone: +421-9610-36999

Office hours:
Monday: 08:00 – 15:00
Tuesday:  08:00 – 15:00
Wednesday: 08:00 – 17:00
Thursday: 08:00 – 15:00
Friday: 08:00 – 14:00