One of the things I struggle with most in doing research is keeping my notes organized. Since research tends to do a lot of branching, keeping notes in a linear fashion seems useless to me. On the other hand, this means that I end up with several notebooks that have ideas and dead-ends everywhere. Then if I want to piece parts together, or if I eventually want to go back and re-investigate what looked like a dead-end at the time -- perhaps because I have learned some new tool -- it takes me a long time to find what I am looking for. How do other people surmount these obstacles?
Whenever I think I've figured something out, I put it in a TeX file. This turns the problem into one of finding the right file among the 1000 tex files on my computer, which is perhaps only a slight improvement. But at least there are tools for this, and I find that if I have to take the time to tex something, I'm likely to remember that I did so.
Of course, this doesn't help for more casual jottings.
Same answer as Charles except I've tried to organize the stuff which is coherent enough to make it to my computer in a personal wiki. I use gitit for various reasons including LaTeX support, but I'm sure there are other options out there. This seems to work pretty well since I can create links between independent but related projects I'm working on.
You should have a look at Dror Bar-Natan's academic pensieve. It's a really impressive effort to live in the future -- it's both an electronic record of virtually all his 'jottings', and an entirely public one as well.
I'm a little less futuristic. I scan all of my notebooks as I finish them (using Berkeley's great high speed PDF scanner), and write an extremely rudimentary index to them. I also number every single page that I write (sometimes the sequence is only approximate, when I have geographically separate notebooks, e.g. office and home). It's great to be able to refer consistently to older sets of notes.
I like using the GNU Emacs Org-mode for note taking. Among the mode's many features are block folding, support for hyperlinks, and support for LaTeX.
An on-line version of the manual is available on the project's web site.
A completely different approach I have (hence a different answer) is to maintain a private wiki... I keep lots of partially written notes on the subjects I'm learning about or on partial approaches in new directions. The advantage is being able to access them from anywhere, without worrying about having a hard copy or the right computer. I organize them in lots of ways (tags, categories, etc.), can search through them easily, and have a history of any changes made. (The site I use is www.wikidot.com)
I have a personal online note-taking system not very different from a private area on the nLab, where I can take all sorts of notes under various headings, and I can add headings/pages etc when I need to, and there is a "glossary" section which is "my own little nLab", as well as a search function, and a separate section for research projects/ideas. There is TeX/MathML support, and since it's online I can use it from work as well as from home which is very convenient. If you really want to know more I could send you a link to the thing (email me) - I am a bit reluctant to make it publicly available but maybe I should just follow Bar-Natan's example and let go. The only real technical problem with going public would be finding a reliable server for hosting it, without having to pay too much.
I would have used nLab for all this if it had existed when I started. Maybe you could try this if you haven't already, and if online note-taking fits your taste.
I'm not always great with organization, but one cool thing I have done is scan my handwritten notes and add them to evernote, which has some really cool handwriting recognition software so I can search through them.
The other thing I do is lots of notebooks with tab dividers, grouped by project.
Well.. for analog .. er.. non-digital notes, they are mostly scribbles of some problem I want to solve..which are all related, because I tend to solve problems that are directly related to my research. I keep two notebooks. I solve a problem in a notebook, and then when I see that my solution got messy.. I move to the other notebook and continue solving maybe browsing at my old solution in the old notebook. So I use one or the other notebook alternatively for trying a problem.. I usually finish the problem in one notebook, but if it gets really complicated the continuation goes on the second notebook.. and I don't try to categorize the problem... I just continue until the notebooks are exhausted then they are archived and I use another set of two new notebooks. Those are my problem and brainstorming notes..
For notes from seminars or whatever else.. I have just one notebook for them all (you dont tend to organize the notes once you reach graduate school and you hardly go to classes). And I usually find what i want in them.. because I dont tend to make notes on seminars talks that I don't understand at all or that I think will not be able help me in the future. And I often take note of keywords rather than the whole seminar talks.
For digital "notes".. they are organized by subject: Algebra, algebraic geoemetry, number theory, topology, ..etc. I often even have folders by names of authors (who are very well known in the field and whose works I often browse) within the say algebra (or whatever) folder. Those are for papers. For mathematical ebooks, I have my own folder I called Mathbooks and I should get organizing them by subtopics as well. I practically live only by digital media.. my 2 notebooks are the only nondigital things I currently depend on in doing research.
I'll echo what Charles said with the addition of using git. You can literally branch your notes just like you would source code and it plays nicely with latex. You can roll back, merge, and share with others very easily. You can tag commits with brief explanations and search those later or use a git visualizer to quickly find something.
Plus all your work is still in text files so you can still use grep and all the usual console tools.
I work on paper notebook with a pencil (I put date here and there as markers). When something is ok I write a small paper (in TeX) with most of the details. Sometimes I publish it. I put everything valuable on my website (published or not). When a notebook is finished I add it to the pile and buy another notebook, pencils are not a problem, I buy them by ten. And, yes, I have also one or two pencil sharpeners.
Works well for me :-)
Whenever I complete anything that might be worth referencing or saving, I type it up in latex, and heap the files in a folder, naming the file with the date I start writing and some keywords. I find that I often have at least a general idea of when I did something, which helps.
Research thoughts are like a tree. Thus linear options like notebooks don't seem to work that well. I would suggest writing all your thoughts on loose sheets and filing them like how you would store a tree in memory. Each parent node keeps a pointer to its children. To mark new nodes in the file, use those colorful stickers that can be stuck on pages which can be seen when the file is closed. These stickers can contain labels, like "new approach 2." Since its a file, you can add extra pages whenever you feel like, and rearrange sections to make it coherent whenever you feel like.
I am a Linux user in an open-source commited university in Caracas, Venezuela, so there is nearly zero limits in the availability of software tools, be it already made or user-defined and honed to my taste, for note taking and organizing. My personal preference in note organization is a no-frills DBMS in MySQL that can be queried online via some small apps in Python or Perl. Granted, as a veteran from the Unix trenches I can say that there is no big surprise for me in the almost harsh way the OS treats the novice user, but even in those cases you can perfectly find a lot of options that offer varying degrees of hand-holding to help you complete your task in less painful ways.
I have been using dokuwiki with a tex plugin for quite a while now to organize my projects, open problems, things I want to read, and a daily research diary. That works quite well and should be usable on most university webspaces (PHP only; no SQL or fancy progams needed on the server).
I use a tablet notebook running Windows 7, so my way is probably not what you're looking for.
I take handwritten notes (my handwriting is rather neat) using Windows Journal. I can save it directly to PDF (if I need to share the notes), or save it in its original format for convenience of editing. Almost all my notes now reside on my hard disk (portability).
I use Mindjet MindManager to create a mindmap of how the notes pertaining to a particular broad topic are related (in content, and time-wise, as further notes are taken). I can easily put in direct (hyper)links to these files right in the mindmap, which is a nice feature since the file is a click away on the map. I can add more notes, web links etc. to the mindmap, as required. I can also link to preprints, journal articles, ebooks (even ebook pages/sections, as I can link to ebook bookmarks on the map) on my notebook, so the note- taking and keeping become quite dynamic, which is nice. With MindConnect, you can also sync the maps online, and even share them with others, or brainstorm with other people on a map online.
I can easily hand-annotate PDF files using PDF Annotator. With a library of ebooks, imagine the web of information you can create right on your computer, in a mindmap. It's quite fantastic. Lots of possibilities there!
I use Jabref to keep notes on particular papers and otherwise keep a single notebook with a table of contents. When it is full I break the spine and scan it. (Looking at the other answers, I see that this is similar to what Scott does.)
For seminars, I just take notes on loose sheets of paper and immediately scan and date them.