Thanks Dave333 for working through this. I was having the same problem. I'd just add a few things for clarification. Log into the old server to access PHPMyAdmin. Navigate to the correct database to find wp_ngg_album, wp_ngg_gallery, and wp_ngg_pictures.
Export them one at a time by making sure that only the chosen one is highlighted, label it and "save as" a .txt file. After downloading the files to your computer (before importing them to the new domain) open them to make sure that the correct name is in the file. Most people will only need to make sure that the new (destination) domain's database be switched with the old one.
(I had additional editing because my files had (wp1_ngg...) - a number one after wp, so I changed them all to wp - even for all of the picture files, but it worked!)
Now log into the new server's PHPMyAdmin. (It's assumed that wordpress has already been installed, so you've already created the new database, and the plugins have been added or replicated.)
Delete the wp_ngg_album, wp_ngg_gallery, and wp_ngg_pictures files, and use the import function to import your .txt files from your computer. (I had to delete them before importing them, but that probably had to do with which boxes I left check marked on the initial export.)
This worked for me, thanks to Dave's input above. Hope that it works for others.