From your brief explanation I understand that your media files are on your external drive. When it isn't connected they won't play. What isn't exactly clear is where the library is, however if it works when the external drive is disconnected then I assume it must be on the internal drive. Typically you can migrate a library by copying a consolidated library (all media is inside the named media folder) from the internal drive to an external drive, move that drive between computers, the copy the library from the external drive to the internal drive and use the shift-start-iTunes method to access the library wherever you have copied it to where it should be independent of the external drive. In you case I think you done things slightly differently, perhaps consolidating the media from the original location to the external drive, copying the .itl file independently, or simply importing the media into a new empty library on the new computer in a way that leaves it connected to the files on the external. Either you need to choose a more appropriate media folder, on the internal drive, and consolidate to it, or you need to revisit the migration. There are detailed instructions in the user tip I linked to in my previous post.
tt2
