Ok..If I understand you now, you have your "site" currently running on phpBB at exigence.dreamhosters.com, and you want it to run on www.exigencersc.com. If i have that correct, to do that you need to either:
1) Move the application files from the exigence.dreamhosters.com directory to the exigencersc.com directory, and modify the configuration files to reflect the new paths/urls.
2) Rename the exigence.dreamhosters.com directory to exigencersc.com, and modify the configuration files to reflect the new paths/urls. (this will kill the exigence.dreamhosters.com site)
3) Use the control panel to redirect/mirror the domains so that visitors going to exignecersc.com see the contents of the dreamhosters.com.site (not the best plan, IMHO).
As to whether any of this will necessitate changes in the site you are currently developing on your own box, that depends on what you are running. At the least you will need to make some changes regarding paths/db connections, etc.
If the "new" site you are developing is going to be what you ultimately want shown at the exigencersc.com site, I'd suggest leaving the existing stuff as it is, and just uploading the new site to exigencersc.com.