If you are not a first world researcher you definitely know what it is. For the rest of us, Sci-Hub is a free (as in beer) repository of scientific articles; articles that would normally cost you upwards of approximately $30 (USD) each, or a slightly less-cost (but still expensive) annual membership with the publisher. Sci-Hub achieves this by accessing these articles with voluntarily (though discreetly) provided proxy accounts that subscribe to these journals and publishers. Once accessed, these articles are available to download from Sci-Hub for all visitors. So naturally it is illegal, because piracy. Thievery is bad. Shame.
More details add wrinkles and grey shades to the picture. The publishers opposing Sci-Hub's activities are registered non-profit organizations but do not function so selflessly. They do not pay for the research in their publications. They do not pay the scientists who critically review and edit these articles for their time and effort. They separately charge submitting authors for printing costs. The also have revenue from ads. So in being a record for scientific data, publishers do not add any real value to the product. However, the system of science has anointed them the gatekeepers and power-brokers of achievement and progress. We have come to unofficially define science as something that is peer-reviewed and published in an established scientific journal. 'X papers in journals with an impact factor of at least Y' is a requisite for most any research position and this dogma gives a lot of power to the middleman publisher (who, I repeat, adds little or no scientific value). Over time, it has created an artificial brand of which the publisher is an under-deserving owner. It is not to say that there is not a sound logic in the concept of scientific publishing and valuation, merely that this brand-based system is extremely vulnerable to greed and outdated in in the internet age.
The central issue with the publisher's monopoly on science is the cost. The publisher owns the window and so, both the producers of scientific information and the consumers have to pay him to access the window. The system has also created an unspoken rule that if you create your own window, it is not as good as the publisher's branded window. The tax-payer who funds the whole machine is unable to access this information that he paid for, unless he pays for it again, which he should not have to. $30 is not a nominal cost, even in the first world, especially considering that the seller pockets all of it and does not pay the producer or even the quality-control crew. If I need to access a conservative average of 60 articles for a report that I am writing, the cost becomes big. Any reasonable research requires access to hundreds of articles and if I belong to a country or system where funds are at a premium, this barrier becomes unbreakable. Local scientific progress is therefore held back because of the cost of brand value. And that is not good. It is this environment that creates and nurtures Robin-hood elements like Sci-Hub. The poor are definitely very happy about it. Sci-Hub has, in short time, revolutionized scientific access in places like India and is of major value to Indian academia.
It could be argued that the poor are just lazy folk that want handouts. The thing is, most of the people accessing scientific data do not intend to sit on it, they will use for science. They are not begging for things they do not deserve, they pay for the research, as tax payers and sometimes as the people that created the data. The publishing company profits big from the whole system. This is not wrong if these profits were directed back to science in some way, by paying for new research. It is, when publishers behave like unbridled, greed fueled, selfish corporations (like you Elsevier). As long as the system of excessively rewarding a publisher who gives back little exists, rebel endeavors like Sci-Hub will keep getting created and celebrated underground.
Am I promoting piracy? No, merely the need for reform in scientific publishing. The onus is on the scientific community to take science back from extreme capitalism. The cost of scientific access should equate to the running cost of publication and no more. There should be less focus on brand value on more on scientific value. The scientific community must actively boycott profit-oriented publishers and use and promote open-access methods. The law must weigh on the wrongs of monopolistic practices by publishers as much as it weighs on Sci-Hub's methods. When a community that prides itself on its intellect and ethical quality is exploited, that, is a shame.