That's the time of the Midnight Sun, so you'll get very long days in southern Norway or even sunshine around the clock in northern Norway. And with the warm weather, there are many things to do and places to go in Norway.
The months of June and July are also the time for Norway's highest tourism levels, so while all sights and attractions will be open, you will see some tourist crowds.
So when should you go to Norway if you want to avoid the high travel season? May and September are wonderful alternatives of when to go to Norway and get lower rates - and the weather in Norway will still be mild enough for outdoor activities and sightseeing.
For travelers, the quietest (and likely cheapest) month in Norway is October. The summer is over, but the ski season hasn't started yet. October can be cold and many outdoor attractions have begun to close, though.
Before May and after September is the year's slowest travel time in Norway, and prices of flights to Norway or local hotels will be at their cheapest levels. While there aren't too many outdoor things to do in the colder months (except maybe viewing the Northern Lights!), this answers a budget traveler's question of when to go to Norway. Just avoid the busier Christmas travel time and the Polar Nights (24-hour darkness a day anywhere above the Artic Circle.)
January and February are dark and the coldest months, so if you're heading to one of the ski areas in Norway, pick March. Now you know what to expect, and it'll be easier for you to decide when to go to Norway.