Right to access in the Nordic countries
The right to access is in some countries the right for anyone to vastly partake of nature and what it offers, as long as it is within the boundaries of local law or general acceptance. Although the Nordic countries is quite expensive, many of the things that are of most value are free-for-all – the scenery, hiking, quite a lot of museums and many other sights.
Not only that, for those who feel like camping, in Finland, Iceland, Norway and Sweden it is generally possible to camp freely in the woods and in the mountains. In these four countries, laws have been enacted guaranteeing "Every Man's Right" to access uncultivated lands, or these rights are considered customary law. In Denmark, with a less sparse population, you do not have the same rights, see Primitive camping in Denmark for some details.
Given knowledge of these laws and camping equipment, it is in fact possible to travel rather inexpensively in the Nordic countries.
Obviously any law that comes with such a right also comes with responsibilities and some limitations. For organized events or commercial activity the rules may be somewhat stricter than for individual hikers. The right to access may of course be limited in nature reserves and the like.
Walking and passing
In these countries, you have the right to walk or ski across uncultivated lands. That means you can walk if there aren't any farmlands or you are not crossing people's gardens. If there are fences, you should look for gates and follow paths, also if there is no apparent farmland (there might be animals, such as sheep or cattle in the area, so always close any gates you open). Also, if there are newly planted trees in an area, you cannot walk through. Other than that, you can pretty much go wherever you like, except areas specifically protected (nature reserves, military areas etc.). Fields may also be crossed using existing paths and when they are covered with snow (and risk no damage). In Iceland cultivated or enclosed areas are exempted and always require landowner's permission (also in Norway the rights are severely restricted in enclosed pastures near the farms, go for the wilderness instead).
Camping and picnicking
Camping is allowed for at least one night in Sweden, one or two nights in Iceland, in Norway two nights in normal countryside and as long as you wish in the wilderness, in Finland "temporarily", which means at least one night and at least two nights if you behave, probably more in the wilderness. You should not camp near houses or farmlands, where "near" means 150 metres in Norway and in all countries far enough that you do not inconvenience anyone and particularly not those in the nearest house. As long as you keep out of the way you should be OK. In Åland you should ask the landowner if possible, but otherwise staying one day and night should be OK. These regulations obviously don't go for those areas especially designated for camping, such as commercial campsites.
If 'out of the way' sounds harsh, it really isn't. There is a lot of free space in Norway, Sweden, Iceland and Finland ...
Pretty much the same rules apply to picnicking.
Rules about camp fires differ between the countries. At least you must not cut any trees or risk wildfire. The camp fire must be built in such a way that the fire does not spread and fire must not be made when forest fires are likely (in practice or according to official warnings, in Finland broadcast with most weather forecasts). Always make sure you have water nearby so you can put out the fire if it starts to spread and any ember when you leave. When you leave, make sure the remnants are cool, also below the surface. Do not make the fire on smooth cliffs, which will crack, or peat, which is hard to extinguish reliably. Leave no trace, unless there is an established fireplace, which you could mend as needed.
In Sweden camp fires are allowed as long as you are careful. In Norway camp fires are not allowed in or near forested areas in the summer. In Finland you need landowner permission, but that permission is granted to the public in large areas of state owned land in the north.
Note that being allowed to make a fire does not mean you can chop trees for firewood. Also leave aesthetically or ecologically valuable dead logs alone. Use twigs on the ground and similar instead.
In many areas the water of streams and lakes can be used even for drinking, at least as boiled. Using private wells requires permission, even when they are far from a house.
Living off the land
It is generally OK to pick wild mushrooms and berries, unless they grow in apparently cultivated areas. In Iceland berry picking is restricted to what you eat immediately, in Denmark to "reasonable", non-commercial picking. Wild nuts can be picked in Norway if you eat them on the spot, in Finland also to take with you.
If you plan to pick mushrooms, buy a good mushroom guide. They are generally quite expensive, but worth their price – there are really nasty mushrooms, even lethal, and some of those resemble edible mushrooms of other parts of the world. There are also poisonous berries, but few that cannot easily be distinguished from edible ones.
Please note: fishing is restricted and hunting always requires licence and permission. Some fishing is allowed without licences, but what fishing varies by country and type of water. Check with the local tourist agency.
Note that picking cloudberry, an orange raspberry-looking Arctic delicacy, is strictly regulated in northern Norway. Picking and eating a few berries is always allowed, though. In Finnish Lapland, commercial cloudberry picking may be restricted, too.
Leave no trace
Whatever you do and wherever you go, it is still your responsibility to leave no trace of your visit. This means you should leave no garbage behind, make sure the camp site you used looks as good when you leave it as when you came. Do not break off any trees. This is the responsibility you get in exchange for the right to access.
More detail
The everybody's rights have a long tradition, but are not well defined. They are mostly based on tradition and on the "what is not forbidden is allowed" principle. In the last decades formalizing the rights has been discussed, but often suggested new explicit law has been found problematic and status quo sufficient.
In Norway, this is a law. You can read either an explanation for travellers or the full text of the law Friluftsloven.
In Sweden, the "Allemansrätten" as it is called, is confirmed in the law but its scope is not formally defined.
In Finland, the "Jokamiehenoikeudet" or "Allemansrätten" is customary law only, except that what plants or plant parts you are allowed to take is defined.
Quirks
Along many popular tourist roads and spots, there are "no camping" signs. These are there to avoid a heavy impact on areas which are particularly popular, and should be respected. Just go a few hundred meters further, make sure you're out of the way, and you're OK.
Although camping vans are OK, it is generally, if usually silently, frowned upon just parking them in a parking area and staying overnight. Obviously mountains and other extreme areas are paved as little as possible. Parking areas are therefore deliberately a scarce resource, and should be used only for parking, not for camping. If you're using a camping van, use paid campsites.
Note that around some cabins in the more popular mountain regions such as Jotunheimen in Norway, extra limitations have been set up which prohibit tenting up to 2 km from the cabins. This is because campers have used sanitary facilities in the cabins without paying. Many Norwegians believe that these limitations are illegal and so blatantly ignore them (and love to be taken to court to have it struck down). They have not yet been tested in court, however, and as a foreigner you might not want to argue about it, so you might want to comply. If you do camp, don't use the facilities of nearby huts without paying the dues. In short: around most cabins you can camp as close as you want (or in a designated area) by paying a small fee – you then also get access to the cabin's facilities. If you don't want to pay, you'll have to go 150 m away. Around some cabins you will have to go even farther away.
In Finland many wilderness huts are free to use, and camping by them (or by other free infrastructure) and using their facilities is recommended. The yards of cabins rented to individual parties are protected like the yard of a home – but Metsähallitus reservation huts are often co-located with open wilderness huts, the facilities and grounds of which can be used also in this case.