Q: my Mac is telling me that my disk is full. I have bought an external hard drive and have just listened to a youtube tutorial on how to move iTunes library to the external hard drive. If I do this can I still load new media on my Mac without the external?

No. Without the external, your iTunes library is not available for update so no new media can be added without it.

Allan

What Allan Eckert says is correct except the majority of web instructions telling you how to "move your library" don't in fact tell you how to move your library, they  tell you how to move  media alone which is not the same.  If your library files (most of what is inside your iTunes folder except the media files) are still on your computer then yes, if you change your media file location back to the internal drive you can download back to the internal drive.  Of course if you have your media folder on the external drive and choose not to turn it on, iTunes will still work but you'll be facing a load of ! because iTunes won't be able to link to the files on the external.  So the bottom line is, yes you can do it, but it won't be easy or pretty so don't.  If you did indeed move the whole iTunes folder ot the external drive then you can still download to the internal drive but you will be using a blank library with default settings because when you start iTunes it won't find hte library file and revert everything to the default location on the internal drive.  You'll then have to restart iTunes with the option key to get it to use the external drive again, followed by going to the media folder on the internal drive and adding that to your real iTunes library.  You can certainly do that if you know what you're doing and want to do it that way.

To do this you need to learn the details of how iTunes works.  For example, you just said, "leave my iTunes on my computer".  iTunes is the application itself so of course it has to be on your computer.  That said, I therefore don't know what you mean when you say leave iTunes on your computer.  The phrasing of your question suggests to me, if I may be frank, you don't understand how iTunes works fully and we can't just give you a "do this" answer and have it work out.  It would be too easy to make a mess of things.

What are the iTunes library files? - http://support.apple.com/kb/HT1660

More on iTunes library files and what they do - http://en.wikipedia.org/wiki/ITunes#Media_management

What are all those iTunes files? - http://www.macworld.com/article/139974/2009/04/itunes_files.html

Where are my iTunes files located? - http://support.apple.com/kb/ht1391

Yes, you can download media and delete it again (but if it is from the iTunes Store there is the possibility you can't always download it again).  Of course if you want to keep a copy you can archive it.  I have done that with some files for which I have no room on my computer.  I transfer them to an external hard drive and delete the reference to the file (but not the file itself) from my library.  If I want a quick listen I go to the file on the drive and use Quicktime player or something else non-iTunes so I don't have to have it added to iTunes and delete again.  I don't know if that is the solution you are looking for.  I could leave the reference in iTunes but if the file can't be located by iTunes it will tell me there is a broken link = !  I guess I could live with that except I periodically scan my library for broken links and don't want it getting confused with ones I have done deliberately and ones suggesting I am really missing a file somewhere.

Another possibility is to make a second iTunes library for material you are archiving and only use that library when the computer is using the external drive.  There are tricky aspects to dong that but it is a possibility too.

The simple solution for most people with computers with tiny drives (as most seem to be these days) is to put their whole iTunes library on an external drive and use it from there.