Direct messages are like mails sent to you directly starting with d twitter id message . With time you will collect lot of direct messages and twitter has no implementation as of now to select all the direct messages and delete it or delete direct messages from a particular user.
However with this Bookmarklet: DM Whacker ( right click and save it as bookmark ) via D Cortesi you can delete all the messages either from a particular user or from one twitter id in a go. This bookmarklet will add a new area just above your twitter id which should look like this.
Once you hit delete it will pop up telling you how many direct messages and sent messages will be deleted. This way you get a count of how many direct and sent messages you had.
I strongly recommend you to bookmark this if you get lot of direct message in twitter and you are a heavy twitter user.