In Windows exist an application named Post-It! Notes. It's a very complete program: it allow to use reminder, show the notes that were opened before shutdown the PC, task manager, ...
Do exist a complete note application for Linux?
Here are a few note taking applications that works on Ubuntu. I believe some of them have task lists and reminders. do not know about the shutdown part but some are extensible so features can be added.
Works on iPhone, Android,Desktop computer or online.
Read more: http://www.6wunderkinder.com/wunderlist/
A fairly good note taking system with image support.
Home page: http://keepnote.org/
Home page: http://www.tiddlywiki.com/
Notecasepro keeps all your notes in one file, security feature, image support. The older version of this, 1.9.8 called notecase, is available in the 10.10 repositories, I do not know if this is the case with newer releases of Ubuntu. you may want to try any one of them.
home page: http://www.notecasepro.com/
If you keep a lot of programming code notes this has some nice preformatting features that are quite useful.
Home page: http://www.giuspen.com/cherrytree/
Emacs Org mode
Very nice Table feature with pure text, good output to Latex, Even spreadsheet calculations can be done. Can be installed from repositories, just search for org-mode in your package manager.
Some tutorials for Org: http://orgmode.org/worg/org-tutorials/index.html
zim desktop wiki
Highly recommended and I must add, my favourite.
Supports linking in place or importing (copy to notes folder)images such as .png, .jpg and other common bitmap formats also supports the vector format .svg, so you can link or import all those .svg drawings you want to reference in your notes. Excellent Latex and Html Export.
With latex system and dvipng installed a Latex equation plugin is availabel to include nice mathematics in your notes.
Home page: http://zim-wiki.org/
Zim in action: http://zim-wiki.org/screenshots.html
Org-mode is an Emacs mode for note taking and much more. It can do everything Tomboy and Gnote can and it can also link notes to a calender and supports export to LaTeX.
First I used Gnome's sticky notes, then I switched to Tomboy, then to Gnote and finally to Org-mode. It takes a little bit more time to learn Org-mode but after you have learnt it you will found it more powerful and efficient.
To install it:
Nevernote is an open-source note taking program. I think it works compulsorily with an Evernote account, however.
There is Gnote also. It's in the Software Center and I hope it fits your needs.
You can use Sticky Notes. See this related question for details: