Functionally URL shorteners take long URLs and convert them into short URLs. They do this first by generating a unique short URL the content of which includes a redirect command to take the user to an underlying long URL that is usually stored in a database. The database typically would simply consist of a table that linked each short URL to its associated long URL.
There are many freely available URL shorteners and some of them are listed below. Note that some services offer added-value features, often for a fee, that let you track traffic or do cleverer things such as redirect traffic to different long URLs depending on where the searcher is located or what time it is. One example of how this can be useful is a shortener that sends traffic from France to a French version of your landing page and traffic from the UK, Canada and the USA to an English language landing page.
URL shorteners tend to have short domain names themselves. Take a look at
Google used to offer a popular URL shortener service at goo.gl but the service was withdrawn in 2018 although existing links are still supported.