+1 for org-mode
While on-topic, just want to clear up misconceptions of org-mode. It is different from all the software/website task managers mentioned here. You can think of it as a meta-organization tool. A org software factory if you will. You can customize it to exactly how you want your thoughts, tasks, notes, etc organized.
See http://doc.norang.ca/org-mode.html for an idea of the sheer breadth of options available to make it work exactly for you.
This may sound like a lot of work, but there are a lot of sensible defaults, which is why most org-mode tutorials that scratch the surface make it seem like a sibling to other task management software. The base case the OP wants would simply be:
S-M-<RET>: create new TODO
C-c C-t D: mark TODO as DONE
If I may pilfer a quote, org-mode "outshines other [task management solutions] in approximately the same way the noonday sun does the stars." (Neal Stephenson on Emacs)