Remository requires the Black Sheep Research Library. Remository will attempt to install the library, or you can download it from https://remository.com/downloads (look in Useful Things) and install it using the Joomla installer. You will need to be registered to download, although a free subscription is sufficient to gain access to the library. Remository itself requires a paid subscription, and can be downloaded from https://remository.com/downloads (look in the Remository folder). It is then installed in the usual way, with the Joomla installer.
Once Remository and the Black Sheep Research Library are installed, they should be kept up to date. The Joomla update system can be used for this. To make the update system work, you need to active the System - Remository Update Authentication plugin (which is automatically installed with the library). Ensure it is activated, and configure it. Go to the "Update Servers" tab and click on the green plus sign to add a server. The first two fields will default correctly, and you should enter your software token in the third field. Your token is shown on the profile page at https://remository.com. You can update by downloading and installing a fresh copy, but it is usually easier to use the update mechanism.

The current version of Remository will work with even very old data. If you are migrating from an old version of Joomla, it is best to remove Remository first of all. Removing it will not delete the database tables. Make sure that your old database tables (jos_downloads_xxx, jos_permissions, jos_assignments - with jos changed to whatever prefix you are currently using) are present in the database for your new system. Then install the current version of Remository. The Remository installer will make any necessary changes to the database tables, and you should be able to use your new system with up to date Remository.
If you used Remository a long time ago, you may have relied on Remository user groups (they were implemented before Joomla had sufficient flexibility to support role based access). If so, it is best to migrate to using the CMS groups. This is a Remository configuration option. BUT before making the change, make sure that you have Joomla groups with the same names as the old Remository groups.