Yes, this is bokmål. Bokmål is used by roughly 85-90 % of the population (according to Wikipedia), including Oslo and nearby areas.
Note that Bokmål and Nynorsk are writing standards, they are not directly applicable to speech.
The grammatical differences are too many to list here. Bokmål and Nynorsk are more different from each other than American English and British English, for example. There are different rules for conjugations, some word endings are avoided in Nynorsk, and so on. Nynorsk is a constructed language, based on dialects.
Personal bias note: I use bokmål and I'm strongly opposed to wasting time learning two writing standards in school, when we could have learned a new language instead.