I have a site I want to backup online, and I have FTP access. Is there a faster way instead of downloading it to my PC and the reuploading it to the backup server?
If you have just FTP access - no shell access - no rsync - then it is not possible to make the servers talk directly to each other. You will need to use your own computer as an intermediate.
If you need to do this regularly then you can script this - for example using the perl Net::FTP module, see this article for an example http://www.linuxjournal.com/article/6686
You could FXP it through your PC if both systems supported it, a good tool to use on Windows to try this is FlashFXP (which is actually a nice FTP client also). If you do not have any sort of other access such as shell or RDP though, you cannot initiate a direct transfer from one server to the other, and in that case, yes you would have to download it, and then re-upload it.
If you have RDP/SSH access to either server, you can push/pull the data directly using a variety of methods.
rsync is the gold standard for this sort of thing, if you have shell/remote desktop access to either server. Lots of options, including the very useful option to only transfer changed files.