In insert mode, Ctrl-o escapes user to do one normal-mode command, and then return to the insert mode. The same effect can be achieved by <ESC>ing to normal mode, doing the single command and then entering back to insert mode. Ctrl-i is simply a <Tab> in insert mode.
In normal mode, Ctrl-o and Ctrl-i jump user through their “jump list”, a list of places where your cursor has been to. The jumplist can be used with the quickfix feature, for example to quickly enter to a line of code containing errors.