A post, or a whole thread? Save it off-forum, or mark it in forum for finding later? Using a forum based method, or something within your internet browser (Chrome, Internet Explorer, Edge, Firefox, Safari, etc)? Saving text only, or pictures as well?
Off- Forum
Posts are easy as text, as Broch describes. It is even possible to use a screen grab program to copy the pictures to paste in so they are there with the post.
Threads are more difficult. You can copy and past them, but usually the format goes all over the place and the result looks nothing like the original thread, even if you have managed to get all the text and photos. You can "print" to PDF, either using something inbuilt to your computer (Windows 10 has this) or something you have downloaded, like CutePDF Writer. This is better than pasting to Word (or other word processor) but will force the thread into paper size chunks, which can break up rather oddly when there are a lot of photos, or they are large.
Saving the thread using Save As will have mixed results depending upon the browser used. I only know about Chrome and Internet Explorer. For saving threads, my experience has been that Chrome is useless. The system is that you get an HTM file and a folder of the same name in which all the images from the thread are stored, the HTM then uses the contents of the folder to make the thread look right. Opening said file in my Chrome and it is a mess, looks nothing like the original. Save As Complete Webpage using IE and the result, when opened in either IE or Chrome, looks like the original, or near enough to make me happy.
Save As in IE also gives the option to save as "Web Archive - Single File" as a .mht file. This gets rid of having to manage a bulky folder full of all the pictures and icons, but the resulting file can only be opened by IE, not by Chrome.
Other browsers may offer different options and have different incompatibilities.
Finally, you can use a browsers in-built system
In Forum
About all you can do is "Watch Thread".
In Browser
You can add the page, or the post, to your Favourites, Bookmarks, or Reading List, depending on what program you are accessing the forum through. This is how I used to do things, back when the internet was young. I had a great list of bookmarks organised by the Forum name that they linked to.
All have advantages and disadvantages, depending on what you want to achieve, and what of the source you are trying to preserve.
Chris