I have written about this subject in my "old blog", but it never hurts to write an updated version with updated misconceptions.
People have all kinds of misconceptions about copyright, such as the following.
"It's not illegal to use it if I don't do it for money."
Whether you are making money from the copyrighted material doesn't matter, it's still copyright infringement, and you could be sued. (The only difference there may be is that the crime might be considered more severe if you made money out of it. However, even if you didn't make any money out of it, you could still be sued for quite a lot.)
"'Let's play' videos do not break copyright."
Actually they do. Most video games contain sound tracks, and unless you play with the audio muted, you are infringing copyright on that ground alone. The graphic assets of the video game are also copyrighted, so using them so extensively in your video could also be seen as infringement.
"But there are tons of "let's play" videos out there and the game developers aren't suing them."
That doesn't matter. Sure, most companies do not care about "let's play" and other similar videos of their games, but that doesn't mean they don't own the rights or that they couldn't sue. In fact, a few companies do send DMCA's and cease&desist demands to "let's play" videos of their games. They have the right to, under the law.
Most companies not doing so doesn't set any kind of precedent. This is also a segue to the next misconception:
"Copyright has to be enforced, or you lose it."
You are confusing copyright with trademarks. Trademarks need to be actively enforced and defended, or you might lose the trademark (in many jurisdictions). Copyright is nothing like that.
"A work has to be registered for copyright."
This might have been the case in some countries in the distant past, but it hasn't been so in a long, long time. Copyright is completely automatic and you don't have to do anything to get it. Publishing your own original work is enough. This is also related to the next misconception:
"If a work does not say it's copyrighted, it's not."
Marking a work with the word "copyright" or the symbol © is completely optional. You own the copyright to your original work without the need to say so or make any kind of markings or reminders. You can't assume a work is not copyrighted simply because it doesn't say so.
"Fan fiction does not break copyright."
In most jurisdictions copyright also covers derivative works, so most if not all fan fiction of existing works infringe copyright. Sure, most authors and companies don't care, but that doesn't change the fact.
Some countries make an exception with derivative works made for the purpose of parody or satire, but what exactly is and isn't considered such might not have been tested in court too many times, so it may not be completely clear.
"This is not derivative work, it's transformative work, which is legal!"
You really don't want to wade in that swamp. The difference between derivative work (which is protected copyright) and transformative work (which is allowed) is really fuzzy, and it depends on the person, the lawyer, the court and the phase of the Moon.
If a big corporation sues you for copyright infringement, are you really ready to try to beat them by arguing that your work is "transformative"? Remember, they have an army of experienced ruthless lawyers, while you probably don't. And in most countries courts tend to side with corporations. So are you going to take the risk?
"But this lawyer's blog post says that this transformative work is legal." He's not your lawyer, and he's not defending you against a megacorporation.
"A performance of a public domain work can't be copyrighted."
Personally this one irks me the most, because it makes no sense, but I simply can't deny facts: An individual performance of a work that's unambiguously in the public domain (because its author died long enough ago) is copyrighted by the performer. You can't use copies of that performance without the author's permission, even though the original work is in the public domain. So no, you can't just download that Beethoven music from YouTube and use it in your video, because that performance will be copyrighted, sorry.
As said, in my opinion this makes no sense (because you are essentially re-copyrighting a PD work, which shouldn't be a possiblity), but it's just how it is, so you should be aware of it.
"If I declare my own work of art as public domain, it has no copyright."
"Public domain" is not a valid usage license. In most jurisdictions copyright is automatic and you can't simply get rid of it. Most jurisdictions do not have the concept of publishing something without copyright (usually with the exception of works published by certain branches of the government).
This being said, you can probably safely use a work that has been declared by its author as being "public domain" without worry of being sued in the future, as courts will probably consider it enough. The work will still have copyright (in other words, it will not actually be in the public domain), but since the author claimed so, most probably no court will allow the author to later take it back. However, this shouldn't be trusted. The work still has copyright, no matter what the author says. And you never know what a random court will decide if you do get sued in the future.
If you want to publish your work with a completely liberal usage license, use a copyright-based license that explicitly states so. There are many existing licenses you can use, such as the BSD and MIT licenses for software and the Creative Common licenses for other works.
Things that are not copyrightable
There are some things that, sometimes perhaps surprisingly, do not fall under copyright. Sometimes there are all kinds of misconceptions about this. Sometimes some corporations egregiously and opportunistically claim copyright on things that are not copyrightable.
Game mechanics, ie. the method by which a game is played, does not fall under copyright. (In some countries these may be patentable, but they unambiguously do not fall under copyright.) For example, if a Russian company tries to bully you into taking down a Tetris clone you have made because that company claims copyright on the mechanics of Tetris, you can safely ignore them. It's not copyrightable.
Individual sudoku puzzles fall under copyright (because they are considered non-trivial original creations), but the rules of how to play sudoku do not. You can safely implement a sudoku game without fear of breaking anybody's copyright, as long as you develop your own original puzzles. (If your game algorithmically creates puzzles on the fly, and it happens by chance to stumble across one that somebody else has already published, it can become fuzzy. I wouldn't be surprised if this has never been tested in court. A lawyer who is an expert on these things may have a more informed opinion on this. I would guess that since your game creates puzzles randomly and not based on anybody's work, it's probably not infringement. After all, you can't control what it comes up with.)
Computer algorithms do not fall under copyright. This is completely and absolutely unambiguous in most jurisdictions. If somebody claims that they have copyrights on a computer algorithm, they do not know what they are talking about (or they are lying). A specific implementation of that algorithm (iow. a piece of source code implementing the algorithm) does fall under copyright, but not the algorithm itself. You can freely implement the algorithm by writing your own original code. (Note, however, that some countries have software patents... They suck, but you can't ignore them if you live in such a country.)
Related to the above: Food recipes do not fall under copyright. Anybody claiming otherwise is wrong. The exact wording, layout and possible images in a recipe book are copyrighted, but not the recipe itself (which is, essentially, a method for doing something). You are free to copy an entire recipe book and publish it as your own as long as you use your own original wording and layout.
Board game (such as chess or go) transcripts do not fall under copyright. You can take the transcript of a famous game and distribute it freely. (Be aware, though, that in this case I'm not 100% sure. I'm perhaps 99% sure.)