Updated Nov, 26th 2012
(This application would really be better for mobile with a texting-like interface, but I'm happy to start with web.)
Sometimes, you want to send messages to someone, but you don't literally want them to see it. Sometimes the person you want to talk to doesn't exist-- or doesn't exist anymore-- or doesn't exist yet. A fictional character. A celebrity. Your ex whom you haven't spoken to in years. A deceased relative. Your future true love. Your future child. A religious leader or a deity. A company that's wronged you. Yourself, in two years, or just when sober again.
When you're bored at work, or lying alone in bed. When you're dying to rant at your boss but can't show it to him, or to sext a friend who doesn't know you're interested... Sometimes, you just want to get it out.
Text Away would let you send messages to illusory recipients, tracking everything said to each person separately. Using an interface as though you're sending an email or social media message, with a profile image for each recipient, it's like journalling in the second person. You can also then read all messages "as" any given recipient, and "write back".
A less barebones feature would be to allow sharing of specific messages (or complete streams) with others, if you do want them to see it later.