Here are instructions and advice for setting up and managing a Cydia repository. For context, Cydia uses an adapted version of Debian APT (Advanced Packaging Tool) to manage packages.
If you have no interest in using your own server, you can use MyRepoSpace, which provides free repository hosting. It's often slow and unreliable though, and it gives you limited flexibility.
The authoritative guide to setting up a Cydia APT repository is saurik's post on the subject.
Patrick Muff wrote "Create your own Cydia Repository on Ubuntu in a few minutes"
WinneonSword made a Tutorial Repository on Github.
Quick and dirty summary
The basic idea is that you have two files in your server, Packages and Release. Packages must be bzipped and named Packages.bz2, and optionally Release may be also. Packages contains all of the information related to the different packages on your server (and where to download them, more on that later) and Release contains all of the information related to your server (like the name, description, etc).
If you want to see examples of Packages and Release files, you can see the cached files from your installed repos at /var/lib/apt/lists (note they can be quite large).
The Packages file mentioned earlier points to .deb files in your server that you can download. These are made with dpkg-deb. Manpage here. The idea is that you set up a folder in the way you'd want the files to appear in your filesystem (and the DEBIAN folder, which would contain your control file, and optional preinst and postinst scripts) when it installs in Cydia, and then you'd use dpkg-deb -b folder_name to make the package (which will be named folder_name.deb). Read more in the Packaging page.
Put the file CydiaIcon.png at the root of your repository. It is displayed at 32x32, and it would be best for the file to be at Retina resolution (64x64 for @2x and 96x96 for @3x).
The easiest way to make your repository "private" is to restrict access based on UDID. Cydia sends the user's UDID via the X-Unique-ID HTTP header, so your server could check that against a database in order to ensure that the user has rightful access.
NOTE: Static webhosts like Neocities won't work for private repos. You'll need a server that has some way to let you process requests server-side, e.g. PHP, node.js, or Flask.
A PHP implementation: A sample UDID-protected Cydia repo by moeseth.
A Node.js implementation: A sample Node UDID-protected Cydia repo by Aehmlo.
Because UDID's can be easily faked, and are sent with every Cydia support email, UDID-protected repos are not very secure. However, you can use a username and password system through Cydia's depiction system, where the user enters their username and password in the depiction page, and then is authorized to download the package. This system also requires server-side processing, so static webhosts (e.g. GitHub pages) will not work.
A PHP implementation: A sample password-protected repo by goeo-.