The short answer is that Popular Front Communism in the United States provides a model of organizing that was effective then and which could be effective today. There are many models of organizing that are available, particularly that of the Spanish Anarchists before and during the Civil War; I'm not claiming that this is in any way more effective than that but instead am offering it as one more tool in the tool box.
So what was the Popular Front? Literally, it referred to a new policy within the international Communist movement put in place after the fall of Germany to the Nazis that emphasized cooperation between Communist parties, Socialist parties, and Liberal parties in order to combat the threat of fascism. In reality, although this strategy was effective in France in electing Leon Blum as prime minister on a popular front ticket, the big advantage of the popular front was that it opened up local parties to be much more independant from Moscow and encouraged innovation in organizing techniques and goals. This was especially true in America, where during this period, under the Communist party leadership of Earl Browder, the size and influence of the Communist party grew to the point where hopeful observers thought it might become a mass party and a permanent fixture in American political life.
The Communist Party at this time was heavily involved with the organization of the CIO, or the Congress of Industrial Organizations, which based itself on the concept of industrial unionism, something that the IWW had pioneered, where industrial unionism meant organizing skilled and unskilled workers in an entire industry, from people working the assembly line to janitors.
The CIO, according to James Ryan's excellant book, used the organizational skills of the Communist Party to help mount campaigns for unionization.
The Communist Party at this time also pioneered civil rights work down in the South, one of the only organizations in the '30s and '40s that was doing this sort of work.
They helped out with Rosevelt's Works Progress Administration on art and theater projects, yes, raised money for the Spanish Civil War, organized support for China--at that time occupied by the Japanese and in any case still in Civil War. Maoism as such had not coalesced yet. Organized Unemployed Councils during the depression, helped elect progressive congressmen like Vito Marcantonio in New York City, who represented a district in Spanish Harlem. In other words they were involved in quite a lot of stuff that would normally be taken as being good.
One of the ways they did this was through fronts. Fronts in this period weren't generally deceptive but were independent organizations that the Communist Party created that were avenues for people to be involved with without actually joining the Party. Additionally, Party members were obligated not only to do Party work but to participate in Front organizations as individuals.
Ideologically, the flexability of the period lead to slogans like "Communism is 21st century Americanism", which was based on a reevaluation of the writings of classical American political authors like Jefferson and Lincoln, with the purpose of finding precedents in their writings concerning social justice that prefigured socialism. This was done in good faith. It was based on the recognition that Communism and Socialism wouldn't get very far if it was portrayed purely as an ideological import from Europe.
The extreme importance of this period comes out when you consider what came directly before and what came after. Before the popular front the doctrine guiding the parties was "Third Period Communism", which was basically pure Stalinism. Third Period communists didn't cooperate whatsoever with other trends and were very ideologically rigid. No surprise then that, unlike in Germany, the American following in Third Period Communism was very, very small. Germany is another story, one that would take us beyond this topic.
Immediately after the Popular Front period, when Browder was deposed for being not enough of a hardliner, the Party did possibly the worst thing it could have under McCarthyism: tell its members to shut up and take the fifth, not explaining their political beliefs and why they participated in these organizations, testimony that could have changed American political life. The theory behind this was that the U.S. was entering a period of imminent Fascism and it would be best to concentrate on building an underground rather than try to sell Communism to the American people at large. Then, after Khruschev's secret speech was published, declaring that, yes, Stalin had committed egregious crimes, there was a mass exodus from the Party after a short battle for political control between the progressives and the hardliners. After this the hardliners came into control of the Communist Party, where they stayed up until very recently...with the difference now being that it's not purely hardliners anymore. I doubt it's a progressive institution. But in practical terms this meant that in the histories and in the propaganda the Popular Front period was demonized as was Browder. In Czechoslovakia after the war, when the USSR was consolidating power over central and Eastern Europe, during one of the rounds of purging the people who were hanged were accused of, among other things, "Browderism".
The main contradiction in the popular front period was that between the relative openness of the local Communist parties and the reality of what was going on in Russia at that time. While local parties were embracing flexable strategies and experimenting with varying the ideology Stalinism still reigned surpreme in the Soviet Union. Not only that but as Stalin's dominance wore on the persecutions continued to get progressively worse. Because of this reality the Popular Front communists were accused of having a secret agenda, of not really in the final analysis being progressive, or, if they really were they were hypocritical in the extreme for taking what was being presented as the reality of Russia issued by Moscow at face value and not questioning it. Interestingly enough, when incontravertable evidence in the form of Khruschev's secret speech came out most of the membership of the Communist Party left. I'll leave it to you to judge the significance of this act in relation to charges of hypocrisy and having a secret agenda.
Nevertheless, during the McCarthy witch hunts a book produced by William Z. Foster, an ultra hardliner who would later become the defacto head of the Communist Party after the mass exodus, produced in the Third Period called "Towards Bolshevik America", was bandied about as 'proof' that the Communists post-Third Period were really not genuine about organizing for social change but were instead conning the public, therefore making them really dangerous, now not just as socialist being part of American society but as infiltrators using front organizations to insidiously push their secret plans.
Beyond the issue of hypocrisy regarding Russia there's also the reality that although people from the International Brigades who went to Spain to fight were genuine idealists, with increased Russian support came other aspects of Russian state organization that weren't quite so positive. The NKVD, predecessor to the KGB, followed the supplies given by the Russians to Spain and went about surveilling Spanish Communist Party members who if suspected of disloyalty were apprehended and tortured and sometimes outright killed by the Soviet Secret police. Documentation for this comes from the book "Stasi", which outlines the evolution of the East German secret police, providing evidence that Mielke, the head of the Stasi, performed this function in Spain after fleeing to Russia to avoid a murder charge against two policemen he assassinated.
So, in summary, a really good time in terms of organizing for social change that could be learned from and applied to today's world.
What happened to the Popular Front? Despite being disbanded, Popular Front politics in Europe lead to the ascendency of the Italian Communist Party, which proved to be the most liberal and most independent Communist Party in Western Europe, actually getting many people elected at all levels and exerting influence in Italy. The French Communist Party benefitted from the Popular Front period two, although their post-war history is that of rejection of those policies and the adoption of hardline orthodox pro-Moscow ways of doing things. The most widespread change that the Popular Front produced was the formation of the Communist state of Yugoslavia, which was largely a product of the grass roots movement of partisan warfare and of basing their support on the people and not on Moscow's influence.
Books: The Earl Browder book is the most comprehensive in respect to these matters. "Being Red" is a personal memoir by the novelist Howard Fast that details his time in the Party, why he joined, and why he believed in it up till the Secret Speech. "If I had a Hammer" by Maurice Isserman details the aftermath of the fallout from the secret speech in that Isserman traces what all the people who left the party did after the left the party. It turns out that they got involved in organizing in the Civil Rights and Anti-war movements, as well as in other things that prefigured the New Left. Isserman's book is really valuable also because it doesn't just focus on ex-Party members but looks at all the different currents of left thought that were present during and after McCarthyism, such as the anti-nuclear movement, the Young People's Socialist League, and very topically the formation of SDS from an affiliate of the Socialist League for Industrial Democracy, an outgrowth of the Socialist Party, to the SDS of the sixties that we know today. Gives actually very, very, detailed accounts of how SDS first formed, what controversies there were between the Socialist Party people who formally had control over it and what the people who were members themselves wanted to do, leading to the cutting of ties between SDS and the Socialist Party, as well as the attempt to co-opt SDS at its start by the Young People's Socialist League, which wasn't successful and which would have basically killed SDS if it had been carried out.