There are already a lot of great answers. If I may add my two cents. Reverse engineering software is akin to the mechanic or tinkerer whom just enjoys taking things apart, understanding how they work, putting them back together, and possibly modifying their subject to adapt its behavior so it is more to their liking. There is no shortage of examples in the physical realm.
Any Electrical Engineer, hardware designer, and the most eliete QA engineers are talented reverse engineers; which is to say, they have an expertise in debugging/analyzing software/hardware implementations. I interned at AMD as I worked my way through school and my debugging skills not only helped me advance to more serious roles within the company, but it is also really fun! Like a game or puzzle that you can play for as long as you want. The challenges will never stop coming.
My favorite example of a legitimate need for reverse engineers is NASA. They are not the only outfit where the phrase "mission critical" applies, where people's lives depend on the quality of their work, but it is the only one that has to fix problems even if their product is hurtling through space. They have to set and freeze product versions years before that software / hardware ever goes to the launch pad. That may seem silly, because the phones in our pocket have more computing power than the entire shuttle, but they can't use modern technology. It's too risky. They have to accept what is available at the time that the specs are frozen and that's it - lest they incur the scheduling and workload that upgrading, testing, fixing, re-testing, verifying again, simulating realistic conditions to determine failure points, fixing it, and starting over.... it is serious business and those engineers do not take their responsibility lightly.
Would you be surprised to learn that NASA projects have used Microsoft OSes on hardware went into shuttle components? It is not much different than using a M$ OS as the foundation for, say for instance, a point-of-sale terminal or for the control system at a power plant. Now consider a scenario where the OS, the hardware, or some portion of third-party software is found to have a flaw in it. Moreover, imagine that the vendor does not care about the flaw to the degree that NASA does and decides that there is no fiscal motivation for them to fix the issue - even if lives depend upon that component not failing. NASA is stuck.
At this point their options are to either replace the component and start their processes all over - a huge cost in time, man-hours, and no guarentee that they won't discover another critical flaw after upgrading; or, instead of upgrading, they can reverse engineer the hardware/software failure and determine if they could fix the problem themselves - maybe it would take some curcuit re-design, changing the values of a few resistors so that voltage levels on the memory bus stayed within required limits under the extreme conditions that were making them fail or maybe there is a binary patch that they can apply that to mitigate the error state they discovered where life-support systems began failing or a probe's batteries drained and left it unable to re-charge itself.
I'm not totally making these scenarios up. They come from stories I have been told, by friends who worked at JPL, my engineering professors when I was in the University, or colleagues that have worked with in the tech industry. All this, and more, has been surmounted by NASA's engineers. It did cost quite a bit more than India paid (tip of the hat to them too), to do what they have done, but the first time through is always more costly.
Technically, NASA engineers would be violating one or more of the intellectual property laws that vendors claim they have (they may or may not actually have those rights; I"m not trying to debate that here). If you bought a corvette, you would be completely within your rights to bore out the cylinders to change your car's performance profile. Different tires. Modified suspension. Tweaked timing and mixture ratios in the fuel injection system. The famous Shelby Cobra is not only modified, but also remarketed with its aftermarket modifications- all entirely legal.
If you bought a radio and wanted to make some modifications, so that it was water proof and you could take it on the lake with you. Amplify it's output so that it would still be audible over the boat engine, or anything of that sort... legal.
Now, jailbreak your iPhone, modify your xBox or the Kinect system that came with it, or, in NASA's case, modify the OS and/or re-engineer the memory bus on a motherboard, to ensure the safety of a human mind you. All of these activities have evoked fierce reactions from the vendors that, for whatever their reasoning, do not want those activities to be continue nor become more widespread. Ultimately, this led to the DMCA laws that we have today being passed. Before that, even if you were clicking "I agree" to the rediculus terms and services that were put before you there was still the question if that contract was legal enforceable.
The DMCA, which does nothing to disuade the massive blackmarkets in China or South America, carries harsh penalties and has been treated as a kind of "hunting license" for those that wish to go after consumers can and have. Should it be illegal to make unlicensed copies of someone elses music; to sell, trade, or even give it away?
Clearly, that should be illegal. Artwork, books, patented processes, protected trademarks; yes, those should be protected too. Software and hardware; again, yes the the author of software should be able to enforce resonalbe aspects of their licensing that protect the the investment that have made to create said software or hardware. Explicitly, it is, and should remain, illegal to puchase one copy legitimately, turn around and make copies, and then sell/trade/or give the unlicensed copies. Microsoft, VMWare, Adobe, and similar big companies should retain that right.
Now, once I have licensed the software or hardware can I make modifications and then sell/trade/or give away the software/hardware that featured my modifications. That depends. It i, and should be illegal, to profit from derivatives works beyond the limit to which you originally licended the art (i.e. software). For example, you could not draw the likeness of a Disney character, change the name, and then claim it as your own and profit from it's use. I believe that Mikey no longer protected, but when he was that was and should be illegal.
Should a computer hobbyist, like the ones that we credit for founding the industry today, be allowed to to explore, tinker, change, discuss, show, or teach any manner of activity that they desire on their legally licensed software and hardware? I won't answer that, but I will say that the DMCA says "no". In fact, the same tools that a reverse engineer would use for debugging the operating system are the very same tools that the prosecution will cite as evidence of illegal activity.
Let's be a little more giving for someone prosecuting a DMCA case where the accused was know to have accessed the cryptographic keys that, among other things, would allow the defendant to make copies of protected art (e.g. music and movies). There are some legitimate reasons that could explain the defendants actions, but not many and the illegal activities that could be pursued are costly to those that are trying to protect themselves. At the end of the day, even though they tried their best, there is no way to actually prevent anyone from accessing those cryptographic keys.
It reminds me of the tag on a mattress waring of insanely harsh penalties for removing said tag, but there is no protection for the tag. Just threats and overreaching penalties that are used to make examples out of as they stoke the fire and declare a witch hunt.
Should Apple be able to prosicute a security research whom respectfully, and sincerely trying to help and support Apple make better products, be prosicuted by the technology giant? Hell no, but you better believe that they tried - that happened. The employees of Target or Home Depot that warned of potential dangers; where are they now? They were fired, run-off, considered a nuisance to the company, shunned, and labeled as "not a team player". In the wake of the data breaches that Target, Home Depot, and countless others (certainly at least one of these events has impacted your life) it is more apparent now than ever that reverse engineering is not only a right, but an aspect of our emerging social structure that is going to be needed more than ever to 1) protect the innocent consumers and digital citizens of the world like yourself and 2) hold accountable those whom are responsible to take reasonable measure to protect the community that entrusted them with their well being (granted in exchange for a service).
The NSAs, the CIAs, the DeutcheBanks, the HealthCare.govs, and the hospitals that we all wanted and empowered to do what they have done. It is the reverse engineer who will become the modern freedom fighter, tomorrows activist, that is going to be on your side, the little guy, when power corrupts and greed lets way to disregard for the responsibilities that were implicit in the power granted to those organizations. To be just a bit more dramatic about it; reverse engineers are the priests of a new religion. We should honor them, applaud their bravery standing up to those drunk on power, and protect them when they risk the privacy of their family to step into the line of fire, on our behalf, because it is "the right thing to do".
If you know one of these individuals- buy them a beer tonight and say thank you. As much as any soldier, they are going to be the ones that protect us when the next digital tragedy is impending.