The menu that shows up depends on the specific plugin you install. The technical details of how it's done can be found in the Codex.
In your example if you installed another contact plugin, lets say the Grunion Contact Form (that's what I have installed on my site at the moment so I know how it works), you would see a new menu listed below Comments called Feedbacks. In the case of the Grunion Contact Form it emails you when someone sends you a message and also stores it in the database so you can see it in the administration area under that new Feedbacks menu.
If you installed both Contact Form 7 and Grunion Contact Form you would see both menus in your administration area. I would recommend you choose one contact form plugin and stick with it for simplicity sake but it certainly would be possible to use multiple plugins simultaneously.
I hope this answered the questions you had - if you want more details about something just reply and I'll do my best to clarify.