Is the bar at the top called "panel"?
No, the top bar is called the "menu bar".
Does the Ubuntu logo at the top left
corner have a name?
It's called the "home button" though it's jokingly referred to as the BFB by developers, for "Big Freaking Button"
Is the left sidebar called "dock"?
It's called the Launcher.
Are the coloured tiles with app icons
on them called "launchers"?
"Launcher icons" or "Launcher items".
Is the purple tile called "workspace
switcher"?
Yes.
Are the fullscreen things that appear
when you click the Ubuntu logo, the
Places tile or the Apps tile, called
"dash"? Do they all have the same
name?
Right; "Dash" is the component name for the overlay that's used for Applications and Places. In non-technical contexts it seems to be used as the Applications Dash and the Places Dash
Other Terms
There seems to be no term decided upon for items kept on the launcher via the "Keep in launcher" function. The relevant dbusmenu functionality is named "pinning_item" in the code; "sticky" doesn't appear anywhere in the wiki and bug reports, and "Favorites" is used in the architecture document once and somewhat ambiguously to refer to what seems to be the "pinned" application items. So "Pinned" and "Favorites" seem both fine for now.
For further information, see the Unity Architecture document, and /Unity/Lenses on the wiki.