The 'Nouvelle Vague' (New Wave) is the term used to describe certain French films and directors of the 1950s and 60s. Influenced (in part) by the Italian Neorealists, they were connected by their desire to challenge cinematic convention. Thus, a radical visual and narrative style was used to deliver content which often (but not always) touched upon the political upheavals of the time. Apart from their movies, the group, which was associated with the film magazine Cahiers du Cinema , is also remembered for their development of what we now know as auteur theory.
As so often, the movement was partly influenced by war. American imports were unavailable during World War 2, so after the war there was a flood of these movies (film noir formed a particularly important influence) which were very influential. Mainstream French cinema itself was regarded by these directors as being somewhat dull and turgid, and they wanted to advance the form with more daring use of camera, narrative, editing and so on. They were heavily influenced by the work of innovative American directors like Hitchcock, John Ford and Nicholas Ray; not only (or even primarily) stylistically, but rather by their perceived individuality and artistry - their status as auteurs.
Chabrol's Le Beau Serge (1958) is generally credited as the first New Wave feature. Truffaut, with The 400 Blows (1959) and Godard, with Breathless (1960) had unexpected international successes, both critical and financial, that turned the world's attention to the activities of the New Wave and enabled the movement to flourish. Francois Truffaut and Jean Luc Godard are probably the best-known of these directors. See also Jacques Demy (for example, The Umbrellas of Cherbourg, wherein he attempts to reinvent the musical in the way that the others did with the gangster movie and the film noir.) Alain Resnais is also very important; see Hiroshima, Mon Amour (1959) and the documentary Night and Fog (1955.)
In terms of style and content, there are a few things to look out for:
Long (VERY long, often) tracking shots (see Godard's Week End from 1967)
Handheld camera shots
External filming as opposed to studio sets (as with the use of handheld shots, this was made easier by the introduction of new, smaller cameras - also, by the very small budgets these directors had)
Jump cuts (see Godard's Breathless from 1960)
Improvised dialogue and plotting
There is a youthful exuberance in the style and content of many of these movies. This is due to a number of factors; not only the desire to reinvent film as an art form, or a new type of art form, but also that many of these directors were originally critics. That is, they knew a lot of film history and theory, but less about the technicalities of film-making. Quite literally in many cases, they were making it up as they went along! As such, 'guerilla' filmmakers owe them a debt. When you're out there, trying to capture a great shot with a cheap camera and no lights, picture yourself as the rightful heirs of this DIY approach!
Contemporary influence is widespread. Tarantino is particularly fond of these directors (Reservoir Dogs is dedicated to Jean-Luc Godard, and the intertextuality of Pulp Fiction owes a lot to films like Godard's Breathless) and his visual and narrative 'innovations' are obviously developments of those made by the New Wave directors (and, to an extent, by the Italian Neorealists some years previously.) Dogme 95 was, in some ways, little more than a contemporary version of the French New Wave, manifesto and all. Equally, the 'New Hollywood' directors were vocal about their influence - see, for example, Arthur Penn's Bonnie and Clyde (1967) The idea of the director as artist or 'auteur' is, of course, massively influential; the whole non-commercial, 'indie' aesthetic, indeed, owes a lot to these directors. Note, incidentally, that there were a number of other national 'New Waves' - including that in Hong Kong. Even films which appeared many years later, like Wong-Kar Wai's Chungking Express (1994), bear the obvious influence of the French New Wave.