Unless the paper is available under a very permissive license, such as Creative Commons Attribution, you will need to seek permission. (There may be other legal possibilities, such as fair use or fair dealing, but that's a little subtle. See this story for more information on that.)
The copyright owner is the person you need permission from. Who that is will generally be marked on the published paper (often it is the publisher, and sometimes the author). If the publisher holds the copyright, then it is still polite to ask permission from the authors as well, although this is not legally required.
Big commercial publishers will often have a department for dealing with this, typically with a name like "Permissions". If you can't find such a department, then you can try just writing to the journal in question (look at their web page to try to find e-mail addresses).
If you are lucky, they will quickly approve your use of the figure. If you are not lucky, they will ask for money.
And, finally, how does the answer vary for (a) those wishing to republish the figure in their own work,
There are definitely legal issues here.
(b) those not wishing to publish the figure e.g. for student coursework.
If you never make the work available to the public, then it is hard to imagine that the copyright owner will ever learn about it or complain (and they would look foolish if they tried to sue someone for using their figure in a homework assignment). However, you still have a moral obligation to cite the source of the figure.