Go to the Apple Icon to left and use the About this Mac item. (Mac OS X version)
Go to the iChat menu > About iChat for iChat version number.
If the App is opening and has no windows (Buddy Lists) then go to the Window menu and see if anything is listed.
It should (possibly) list your Logged in accounts and list Keystrokes next to them.
If there are no Logged in accounts then go to the iChat Menu > Preferences > Accounts and tick the Enable this Account box(es).
If you use the red blob top left of any Buddy List this only Hides the window (list) and does not close/Quit the App.
When you restart the App the windows remain "Hidden"
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.4)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously