To see the Messages sent to or by you, go to Menu > Messages.
Each message thread between you and a specific Member, group of Members, or the Community Managers will appear in this list. Click the Message to see the history.
You can continue the conversation by typing into Send a message and clicking Send. You can also click Attach a Post to add a post to the thread.
You can search through the message thread by typing your keyword into the Search this Message box in the upper right hand corner. The Gear icon allows you to either Archive the message (there is also an Archive Message button at the bottom), or Mark it as Unread.
Note: In some communities, you can only send messages to community managers. There are communities where messaging is disabled.