One common way is to create a sub-domain such as test.yourdomain.com or staging.yourdomain.com and develop the new site there.
When you create the sub-domain, it will make life simpler later if you use the same ftp user for the new sub-domain as you did with your previous site. There are some simple steps you can find here in the forum to rename directories to publish the site.
If you will be starting fresh you can simply make a new database for the new install. If on the other hand you want to migrate currrent content etc you might want to make a copy of the existing database and use that for the new install OR use the same database with a different prefix. In either of these case, the "later" picture becomes more complicated but that is outside the scope of what you have asked so far