You can't copy-paste between Emacs and a shell without using a system-wide copy-paste facility. Text terminals in general do not provide a copy-paste facility.
It seems that you're running this terminal Emacs in a terminal window in a window environment. Window environments do provide copy-paste; that's what happens when you use the mouse. However, this happens outside the knowledge of Emacs and Bash: what you copy is the text that appears on the screen, and what you paste appears to the program as if you'd typed it very quickly.
If you want to integrate Bash's clipboard with the window environment's clipboard, you can use command line copy-paste tools: xsel or xclip on X11,
pbpaste on OS X,
/dev/clipboard on Cygwin. See Share the clipboard between bash and X11 for the Bash side. You can do something similar on the Emacs side, though it doesn't make that much sense since you can run a window Emacs.
If no window environment is available, you can run both Bash and Emacs inside a terminal multiplexer such as Screen or tmux. Their copy-paste facility suffers the same limitation as the mouse-based one: since it's provided by the terminal, it can only copy output that's on the terminal and provide input as if it came from the terminal.
If you want to exchange data between Emacs and Bash, and no common clipboard is available (e.g. because the programs are running on a remote machine), use a temporary file.
Another possibility is to run your shell within Emacs, as suggested by DoMiNeLa10
. In that case, Emacs provides the copy-paste facility.