Thick books has been written about Notre Dame and there's a lot of history connected with the building. Notra Dame is perhaps one of the most well known buildings in the world, and that is not without reason.
It is an impressive construction. The architecture is unique in the world and many has used it as inspiration. From the outside you can enjoy all the fascinating and amazing details. Saints and kings, prophets and religious figures, devils and gargoyles. The details are overwhelming and even if you don't feel like lining up to come inside, it's worth a visit just to see it from the outside.
Once you've entered you should remember that it is a cathedral and place of worship. Since it's a catholic church, you'll find a lot of small altars and niches with saints and often find people in pray before an icon with lit candles. In several places you'll find small boxes where you can buy candles, which you then can light and place before the saint of your choice. Jesus and Virgin Mary are the two most visited, but you'll find that other more local saints will have candles lit before them as well.
A good piece of advice is to show respect for the worshippers. Cover shoulders and legs, don't wear shorts and short skirts. Many French women still wear a scarf and long skirts when attending mass on Sundays. Speak softly and walk calmly and quietly while absorbing all the impressions. There's a unique mood in the building, which is destroyed somewhat by the many tourists.
There are masses several times a day every single day, though most on Sundays. More information can be found at the official website: http://www.notredamedeparis.fr/
Notre Dame is open all days from 8:00 am to 6:45 pm, though until 7:15 pm on Sundays and holidays.
Since it's a functioning place of worship access is free.