France's nationalist wave
To start with, there's the designation "far right" applied to the National Front, including in the headline. "Far right" is often used, more out of laziness than malice in most cases, to conflate several European political movements which have a few things in common but are fundamentally different. Sane, legitimate, and broadly-popular groups like Britain's UK Independence Party and the Dutch Freedom Party are lumped in with genuinely dangerous extremists like Greece's Golden Dawn or the British National Party. These groups are not the same, and if you don't grasp the differences, you won't understand what's happening in Europe and will continue to happen in the future.
France's National Front is a special case, having actually moved from one category to the other. Originally, under the leadership of Jean-Marie Le Pen, it did indeed merit its image as a far-right, crypto-fascist party. The founders included nostalgics for the Nazi-collaborationist Vichy regime and other neo-Nazi elements, and the group emphasized a rigid version of the local religion (Catholicism) and repeatedly indulged in anti-Semitic, anti-gay, and anti-feminist rhetoric. None of these stances is popular in France, and what electoral success the Front had was due to its opposition to Muslim immigration, an issue the mainstream parties have failed to address (this pattern recurs in many western European countries). This was ironic in that the Front's anti-Semitic, anti-gay, and anti-feminist views put it on the same side as the militant Islamists, who are the main advocates of such attitudes in western Europe today.
In early 2011 Jean-Marie Le Pen retired and his daughter Marine took over the party leadership, and took the Front in a different direction. The opposition to immigration remained, but the Front made peace with secularism and modernity, purged anti-Semites, and shifted its focus to opposing the European Union and its overbearing incursions into the sovereignty of its member states, including France.
(A trick that can help you tell the popular nationalists from the fringe extremists: look at pictures of their public marches and rallies. If the crowd is all-male and no one is smiling, it's an extremist group, usually either fascist or Islamist.)
The Huffington Post mentions a few specific stances of this "far-right" party -- protectionist, anti-EU, anti-immigrant. Let's look at these.
Protectionist: It's traditionally been rightists who tore down trade barriers and exposed workers in developed countries to competition from low-wage Third World countries. It's been the left that denounced the disastrous real-world consequences of this -- falling wages and loss of industrial jobs -- even if mainstream leftist parties such as our Democrats have been maddeningly unwilling to act accordingly. Protectionism is not "far right".
Anti-EU: The EU has wrecked the economies of southern Europe, partly by introducing the euro currency, but mainly by imposing austerity policies which prioritize deficit reduction over growth in spite of sky-high unemployment -- the same policies that Republicans in the US advocate. Socialist parties, such as the one currently in power in France, have been punished by the voters because the EU does not allow them to govern as socialists. Opposing the EU is not "far right".
Anti-immigrant: The immigration issue in Europe does not resemble the issue in the US. The main differences are that (a) European countries do not have a long history of high immigration as the US does, and are much more densely populated than the US, and (b) their immigration has come largely from Muslim countries, introducing a militant and sometimes violent religious element into societies far more secular than the US. As I've discussed earlier, militant Islamists are western Europe's closest equivalent to our Christian Right. Opponents of Islam are routinely denounced as racist, and there are some racists who use anti-Islam rhetoric as code, but opposing militant Islam itself is no more racist than opposing the Christian Right is.
France has by far the largest Muslim population in western Europe, so the issue is especially hot there. Most immigrants and their descendants are assimilating, but a militant hard core remains, noisily agitating against secularism, gays, Jews, free expression, and modernity in general, and sometimes even plotting terrorist attacks, though nowadays these are usually caught by the police before being carried out.
Mainstream parties and media have responded to popular anger about austerity, the EU, Islamism, etc. mostly with stonewalling and name-calling. Pim Fortuyn, the founder of the first Dutch anti-Islamist party, was a gay, Marxist professor and quickly became extremely popular, but that did not stop the media from branding him "far right".
This being the case, it's hardly surprising that exasperated voters increasingly support explicit nationalists like the National Front or Britain's UKIP, who are willing to address the real issues (and when no such party is available, there are sometimes significant protest votes for dangerous extremists like the pre-2011 National Front). The trend will continue to escalate until the mainstream parties change their attitude. As long as Americans lump together disparate groups using buzzwords like "far right", and try to understand Europe in terms of American political concepts which don't apply to a very different set of issues, they will not understand what is happening there.