The person who fails to reply in a conversation, therefore dropping that conversation and rendering it DEAD. This can be through several means: e-mail, texting, IM-ing, posting a message on someone's board, writing letters, or any channel of communication that requires a response from both participants after some slightly delayed period of time.
If a person is notorious as a convo dropper, some people may stop messaging him or her as much because nobody likes to start a conversation that will just end up dying when it reaches that person's phone, house, or computer.
Guy: "My girlfriend. Every time we start texting, she eventually just gets bored and stops replying."
Friend: "Man, you have to be the convo dropper every once in a while. That'll prove that you're not a chump"