Thanks a lot, ifmn and Zlobomir!
I think I know what I've got to do now, but there's just a couple more things that I'm still a bit unsure about. ifmn, you know when you're saying to just take a copy of the ApexDC++ folder (i.e. C:\Program Files\ApexDC++) and put this in the same place on my new computer? Do I then have to re-install ApexDC++ on my new computer after copying this folder on to it, or is it just a simple matter of setting a shortcut on my Desktop to ApexDC++'s ".exe" file (i.e. the file, ApexDC.exe that's in the ApexDC++ folder) to get it to run on my new computer? There's no files that are put in the Windows folder as well when you install ApexDC++, are there?
Also, what do I have to do re changing the Certificates and Logs folders? Sorry, I'm not sure what you mean about changing these folders, ifmn. I've just had a look in these, and I see they're both empty. I couldn't see the Certificates folder at first, but I've now found it as a sub-folder of the Settings folder, with both this folder and the Logs folder being sub-folders of the ApexDC++ folder.