Digitalarkivet – Arkivverket (National Archives of Norway) (Norwegian and English)
The National Archives of Norway is the most important website for Norwegian research. Among the many resources freely available on the site are: Parish registers, Census and electoral rolls, Property records, Probate records, Emigration records, Court records, and Tax records.
Registreringssentral For Historiske Data (Norwegian Historical Data Center) (Norwegian and English)
The Norwegian Historical Data Centre is transcribing and indexing Norwegian census since 1865 and some parish records. If you do not find what you expect in the index, it likely means that the record you are looking for has not yet been indexed.
This site contains a significant amount of information on Norwegian emigration, focusing on the ships (including pictures), agents, and steamship lines. The Norway Heritage Project includes an index of passenger lists from 1825-1873.
FamilySearch, in addition to having indexes to parish records and 1875 census, it has a significant amount of information on how to research in the Norwegian records in the Wiki and Learning Center.
Ancestry.com has indexes to parish records (N.B. that while the database descriptions point to the FamilySearch collections they are not exactly the same and you should check both indexes) and for determining the place of origin in Norway they have the Archives of the Evangelical Lutheran Church in America.
General Information on Norwegian Research
DIS-Norge Slekt og Data (Norwegian and some English)
DIS-Norge is a very significant site for Norwegian research. It is primarily in Norwegian but parts have been translated. There are web forums for asking questions (you can ask in English and people will usually respond in English).
The site also has the very important index of grave markers (Gravminner i Norge) and they are working on indexing parish records. Also, if you join you can submit a GEDCOM of your research in DIStreff and find if other people working on the same lines. Its a great way to connect to Norwegian relatives.
Oluf Rygh: Norske Gaardnavne (Oluf Rygh: Norwegian Farm Names) (Norwegian and English)
A database of Norwegian farm names from volumes 1-17 of Oluf Rygh’s almost definitive series. If you know the farm name your ancestor came from this is likely the first place you should check.
At St. Olaf College, they have a great collection of records related to Norwegian immigrants. Online they have a nearly definitive bibliography of Norwegian-American newspapers and locations where they can be found.