Two different passwords
A PDF can be protected in two ways, and the difference decides whether the protection can be lifted at all.
The open password encrypts the file. Without the key the content is unreadable, for anyone. Whoever does not know it does not get in, and that is how it should be.
The permissions password leaves the document open but forbids printing, copying or editing. The file is not truly encrypted here, the restriction is more of a note that programs are expected to respect.
What can be lifted
Permissions can be removed with Unlock PDF when the document is yours or you received it legitimately. Typical case: an invoice that refuses to print, or a bank statement you cannot copy from.
For an open password you need the key. Entering it and removing the protection afterwards is possible, working around it is not. Anyone promising otherwise is promising too much.
The legal side
Lifting the copy protection on a document you own is unproblematic. With someone else's documents it looks different, especially where the protection covers copyrighted work. When in doubt, ask whoever the file came from.
The other direction: protecting your own
If you send a document not meant for everyone, Protect PDF sets a password. One note on that: do not send the key in the same email as the file. A second channel, a message to their phone for instance, is what makes the protection worth anything.
Telling which protection you are facing
The difference shows on opening. If the program asks for a key straight away, it is an open password. If the document opens and only printing or selecting does nothing, it is a permissions password. Many viewers also list under document properties what is allowed and what is not.
Where this comes up day to day
Bank statements and invoices. Many banks and utilities restrict their PDFs. Anyone wanting to move the amounts into a spreadsheet is blocked by disabled copying. After unlocking it works, and PDF tables to Excel takes tables further directly.
Official forms. Some are protected to the point where fields cannot be filled in. Fill PDF forms usually manages anyway.
Your own old documents. A file protected years ago whose key nobody remembers. If it was a permissions password you can get further. If it was an open password, unfortunately not.
Common questions
Is the file altered when unlocking? The content stays, only the protection settings fall away. Text, images and page count are identical afterwards.
How secure is a PDF password anyway? With current encryption and a long key it is serious protection. With a short key it is worth little. Anyone sending genuinely confidential material should choose at least twelve characters, a password generator helps.
Can I unlock several files at once? Yes, provided the same applies to all of them.
What to consider when passing it on
An unlocked document still carries its metadata: author, program and creation date. If the file goes outside, that is worth a look and possibly a round of removing the metadata.