dubmdell, Swedish is a descendant of old Norse, just like Norwegian, Danish, Icelandic and Faroese. Danish and Swedish lost the case system and the three genders gradually from the 16th to the 20th century. There are still remnants of old dative forms and genders etc., but they're not productive anymore. One example is that when we ask the time in Swedish, we ask "What is the clock?", and the reply can be "She's 11:30". The clock is still considered feminine in standard Swedish by many speakers, just as a remnant of the old gender system. We do have two noun classes still, though. The masculine and feminine nouns merged into a class we call "utrum", while the neutral nouns remained "neutrum". Any m/f nouns take the indefinite article "en", while neutral nouns take "ett". This is also the case in Danish and in Bokmål, the most common written form of Norwegian.
Where it gets complicated though, is the isolation of communities in Sweden and Norway. Many dialects in rural Sweden and Norway retains features that disappeared from the prestige language of the cities. Norwegian is even more complicated, since genetically it belongs to the West Scandinavian group of descendants from old Norse, while Danish and Swedish form an eastern group. However, since Norway was ruled by Denmark for hundreds of years, they adopted written Danish as the standard writing for Norwegian. The descendant of this Danish is called Bokmål, it is very similar to written Danish and is used by 90% of the inhabitants of Norway. The spoken language does not correspond to the written language, and is divided into three dialectal regions, the east (containing Oslo), the west and the north. Eastern Norwegian has adapted to Bokmål, and sounds very similar to Swedish, and I as a Swede could understand it perfectly without ever studying it. Almost all who speak eastern dialects write in Bokmål. In the west and north, however, the language has not adapted as much to the Danish-based Bokmål, and some pockets of dialect are more similar to Icelandic than Swedish or Danish, or completely unique. Thus in the 19th century, when Norwegian nationalism was rising, a man named Ivar Aasen went around the country to collect examples of dialects. He compiled a grammar and a lexicon that was a compromise between the many dialects of western and northern Norway. This was the base for New Norwegian, Nynorsk, a written language based on rural dialects instead of Danish. It has been refined and standardised since then, and is now used by around 10% of the inhabitants as their written language. As a compromise, it's not identical to any of the dialects, but can be used with slight variations by any speaker of western or northern Norwegian. It's also completely possible to speak a western dialect and write in Bokmål. The written and spoken Norwegian are separate.
I'll give you an example from a friend of mine who worked in a call center with me.
In bokmål, "How can we help you?" is spelled "Hvordan kan vi hjelpe dere?" and pronounced similarly in eastern dialects. My friend from Stavanger in western Norway pronounced this "Korss'n kan me helpa dokke?" and wrote it "Korleis kan me helpa dykk?" in Nynorsk. He also used three distinct genders, "ein, ei, eit" as the indefinite articles for masculine, feminine and neutral respectively, which is also a feature of Nynorsk, instead of "en, et" in Bokmål.