Apartments typically come with a phone capable of making national and international calls. The bill for this must be paid monthly at the closest Post Office. It is likely to be connected with Romtelecom, the previous monopoly operator.
Fees depend on the provider with all of the rates varying. To get a line installed you need to have a contract with the provider you choose. The time to get the landline installed ranges from a couple days to a couple of weeks. Telephone cards can be used to make international or national calls from all public phones. These can be bought at various shops such as kiosks and newspaper shops.
Vodafone offers a joint fixed and mobile service with information in English here:
http://www.vodafone.ro/english-summary/living-in-romania/
Romtelecom has some information in English here:
http://www.romtelecom.ro/en/index.html