I noticed you can move the chat frame around in The Old Republic, however I can't figure out how to move other elements around such as the local map, party frames and the like. Is it possible to reposition these elements?
In the most recent patch they have allowed you full customization of the UI.
Players can now customize their User Interface by moving and resizing UI elements!
You can literally move everything that is displayed on the screen and resize/customize it all.
Here is a youtube video that shows it in action:
As Sorean mentioned the operations frames can be moved but it's not obvious how to make the party frames movable.
You must convert the party frames to operations frames and those can be moved / resized.
Open your Preferences, select User Interface, and check Use Operations Frames as Party Frames.
You can also change the height and width of the healthbars for the frame here.
Once that is done you can right-click the little dongle in the upper left corner of the Operations Frame to Lock / Unlock. When unlocked you can drag the frame around the screen.
Info and pics from MMO Site
Also a video tutorial of the steps is available at TORHangout on Youtube