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Jean-Luc Mélenchon (b. 1951) is a French politician, former Trotskyist. He considers himself as a republican socialist or an ecosocialist (though his platform is broadly social democratic). Some people who aren't Mélenchon — whether they support him or not — generally consider him a social democrat instead.
Early Political Career
Before starting his own political party, Melenchon was a socialist for 30 years, even becoming at one point the youngest ever French senator and serving as Minister for Vocational Education from 2000-2002 under Prime Minister Lionel Jospin. Inside the party, Melanchon was a left-wing maverick as well, being the main opposition to a "Yes" vote from inside the Socialist party to the 2005 referendum on the European constitution, going against future President Francois Hollande and the rest of the party leadership. ("No" ultimately won by 55%)
In 2008, Melenchon decided he had enough, however, and moved to create his own party, the Left Party.
In 2012, Melenchon ran for President, winning just over 11% of the vote in the first round.
Following the unsuccessful Presidency of Francois Hollande, who has refused to run for a second term, and the near collapse of the Socialist Party as a result, Melenchon, who was now leading the Unsubmissive France coalition, a big-tent coalition made up of members from the Greens, the still-powerful French Communist party, his own Left Party and others, snatched up much of the left-wing vote and steadily rose in the polls, even appearing to be a potential contender that could seriously rival Le Pen or Macron.
Melenchon wants a large 275 billion euro Keynesian stimulus package over the course of 5 years, from mid-2017 to mid-2022, with the goal to reduce unemployment from its current level of 10% to 6% by then. Of this 275 billion euros, 50 billion would go to protecting the environment, with half of that going towards funding a transition to renewable energy, 45 billion would go to social projects, of which 18 billion would go to developing new housing, 102 billion would be invested towards better public services, 33 billion would towards fighting poverty, 32 billion would go lowering the retirement age to 60 and the rest will go to various cultural and educational programs.
How would he pay for it? Melenchon plans on levying a 90% tax rate on income of 400,000 euros, higher taxes on property sales and luxury goods, taxing robots, closing various tax deductions/exemptions created by Hollande (which would be in part funneled towards a corporate tax cut from the average rate of 33% to 25%), clamping down on tax evasion and radically altering the rules of the European Central Bank, ending its independence and allowing the government to borrow from it at 1% interest from it, as banks can currently do, in addition to targeting a higher 4% inflation rate and forcing a devaluation of the euro vis-a-vis with the dollar.
Additionally, Melenchon plans to boost labor rights, enforcing a 35-hour working week and boosting the minimum wage by 16%. 
All good, bar the tax on robots - lowering the amount of labor inputs through lower retirement ages and less working hours means more capital (ie robots) are needed to do the work for us to maintain a higher standard of living, something that a tax on capital would prevent, which is especially counterproductive in the context of falling French productivity for years.
Melenchon is fairly left-wing on the topic of France's involvement in the world too, advocating for a withdrawal from NATO and the adoption of a non-aligned position, critical of the west's involvement in Ukraine and Syria, arguing against war and in favor of Russian involvement in Syria as a way to exterminate Daesh. Additionally, Melenchon advocates for a renegotiation of France's commitments to the EU or else he'll hold a referendum on the topic of whether to leave, namely due to opposition towards neoliberalism and his vision of a free, independent France.
Melenchon is a liberal, supporting legalization of cannabis and euthanasia, railing against police brutality, arguing for improving the conditions of refugees in France (During the debates, he mocked conservative candidate Francois Fillon for arguing for the reinstatement of quotas, saying that no matter what, immigrants will still come in) and criticizing racism against Muslims, saying "Jews were persecuted, then Protestants, and today Muslims" in context of the Burkini Ban scandal in France in 2016 as well being an outspoken feminist, arguing for laws to affirm the equality of women in government, work and society, strongly supporting abortion rights and calling for an end to prostitution and the degradation of women.
Melenchon is a staunch enviromentalist, arguing for a progressive water tax to discourage overuse, the swift transition towards renewable energy from fossil fuels and the eventual shutting down of France's nuclear plants. A big supporter of animal rights too, Melenchon himself as a vegan and argued during an interview about the imperative to reduce consumption of meat on moral and enviromental grounds, even adding the goal of stopping animal abuse to his campaign platform.
Abolition of the French Fifth Republic
In his arguably most radical proposal yet, Melenchon desires a total revamp of France's government, planning to be the last President of France before a total transition to a parlimentary system without the upper house, in addition to enshrining in the constitution other major changes, such as lowering the voting age to 16, mandatory voting, the ability to propose laws and recall MPs via referendum, tight restrictions on government officials going to work in related sectors in the private industry and a ban on lobbying.
In the end, Melenchon only snapped up 19.6% of the vote, putting him behind Francois Fillon, Marine Le Pen and Emmanuel Macron. That being said, if Benoit Hamon, the socialist party candidate, teamed up with Melanchon, as was considered, their combined vote would be a leading 26%, 2 percentage points above what Macron got, allowing them to advance to the 2nd round against Macron in a firm position.
After news of his defeat, Melenchon refused to endorse Macron but advised voters to not make the "Terrible mistake" of voting for Le Pen, suggesting that Macron appeal to disaffected voters by shelving his labor reform plans.
After the 2017 Election
In the following 2017 legislative elections, Melenchon won a seat representing the 4th constituency of Bouches-du-Rhône department (Marseille) in the national assembly. Given the Republican's implicit alliance with Macron and Le Pen and the National Front's implosion, Melenchon has become the de-facto opposition to Macron, mostly rallying unions to protest Macron's labor reforms. Despite this, Melenchon and his party's polling numbers have dropped since the election and whether or not this strategy will be fruitful in the long-run remains to be seen. 
- https://www.theguardian.com/world/2012/apr/06/jean-luc-melenchon-campaign-interview Jean-Luc Mélenchon: the poetry-loving pitbull galvanising the French elections
- https://www.aljazeera.com/news/2017/04/profile-jean-luc-melenchon-170402123518317.html Profile: Who is Jean-Luc Melanchon?