There's something on the internet called "Godwin's Law", which basically says that if an argument goes on long enough, someone will eventually bring up Hitler and/or the Nazis. Similarly, in any discussion involving limitless free energy, someone will eventually bring up Nikola Tesla - and probably the Nazis as well (to describe you, when you tell them what idiots they are).
First, let's examine who Nikola Tesla really was - in the real world, where we all live. Tesla was one of the most brilliant and iconoclastic inventors of the late 19th and early 20th century. He was born in the Austrian Empire in 1856 in what is now Croatia; however, he's ethnically Serbian, which is why he's on modern day Serbia's 100 dinar bill.
His research focused on electricity - specifically high voltage and high frequency electricity. His most significant single contribution is arguably the realization that alternating current "AC" is a much more efficient way to distribute electrical power than direct current "DC", which was favored by Edison. This led to a protracted argument with Edison, which Tesla eventually won. The key victory was when Tesla and Westinghouse won the contract to supply the lighting for the Columbian Exposition in Chicago in 1893.
A fight he never really won was his fight to get credit for the invention of radio. Tesla lectured about the possibility of using electromagnetic waves to transmit information as far back as 1893, and he designed a remote-controlled boat in 1898, which he tried to sell to the government. In 1900, Tesla patented a "system of transmitting electrical energy" and "an electrical transmitter". When Marconi claimed to have "invented" radio in 1901, Tesla challenged him in court. The courts issued a series of contradictory rulings, but whatever the legal standing, history remembers Marconi rather than Tesla as the father of radio. This arguably helped trigger the bitterness that characterized Tesla's later life.
Tesla was obsessed with the idea of transmitting power using high frequency electromagnetic waves, and he designed and built an enormous tower to do so, called the "Wardenclyffe Tower" to do so. There's nothing impossible or mysterious about this concept. It's just that it's not economically viable. Imagine if you had a bar that served beer by spraying it out of a sprinkler and then you would drink whatever you could catch in your glass. It would be a lot of fun, but the bar could only make money if they either got their beer for free or the charge one hell of a lot for the glass. It's the same way with transmitting power. Most of the power would simply radiate away into space. It would only work if electricity were too cheap to bother metering.
Tesla had other ideas for for energy, none of which would have worked. In 1901, Tesla filed a patent for an "Apparatus for the utilization of radiant energy". This was simply an antenna designed to capture ambient electromagnetic energy. The problem is that this energy doesn't exist at a level that would be useful in any way. Nevertheless, generations of free energy enthusiasts have latched onto this as one of Tesla's many contributions that have been lost or were stolen from us. Of course, it hasn't gone anywhere. All the drawings are still there at the patent office. They simply don't work.
Tesla's ideas go beyond free energy. Late in life, he proposed the idea of a particle beam weapon, which he referred to as "Teleforce". Like his free energy device, all the drawing still exist, and also like the free energy device, it simply doesn't work.
Adding to Tesla's mystique is the fact that upon his death, the FBI order the seizure of all of his personal papers. It's likely that the FBI believed Tesla's claims about his particle beam weapons, and seized his papers in the interest of national security. This has fueled decades of speculation about what was in the writings; however, given the nature of the things that are known, it's highly unlikely there were any major secrets in the lost material.
The fact is, everything Tesla did is well understood - not only in light of modern physics, but in terms of physics at the time. In contrast, his more outlandish claims would require a significant modification of physics as we understand it.
It's unfortunate that someone who made as many genuine contributions to our lives as Nikola Tesla is instead remembered by so many for his more outrageous and unsubstantiated claims.