If you want to move the repository and keep history, you’ll probably need filesystem access on both hosts. The simplest solution, if your backend is FSFS (the default on recent versions), is to make a filesystem copy of the entire repository folder.
If you have a Berkley DB backend, if you’re not sure of what your backend is, or if you’re changing SVN version numbers, you’re going to want to use svnadmin to dump your old repository and load it into your new repository. Using svnadmin dump will give you a single file backup that you can copy to the new system. Then you can create the new (empty) repository and use svnadmin load, which will essentially replay all the commits along with its metadata (author, timestamp, etc).
You can read more about the dump/load process here:
http://svnbook.red-bean.com/en/1.8/svn.reposadmin.maint.html#svn.reposadmin.maint.migrate
Also, if you do svnadmin load, make sure you use the --force-uuid option, or otherwise people are going to have problems switching to the new repository. Subversion uses a UUID to identify the repository internally, and it won’t let you switch a working copy to a different repository.
If you don’t have filesystem access, there may be other third party options out there (or you can write something) to help you migrate: essentially you’d have to use the svn log to replay each revision on the new repository, and then fix up the metadata afterwards. You’ll need the pre-revprop-change and post-revprop-change hook scripts in place to do this, which sort of assumes filesystem access, so YMMV. Or, if you don’t want to keep the history, you can use your working copy to import into the new repository. But hopefully this isn’t the case.