About the Operating Systems
An operating system is a collection of software that enables you to use your phone and run other applications on it. An alternative Operating System that is based on Android and is made to replace Android is called a "Custom ROM". Meanwhile there are countless Custom-ROMs out there, some of which have a particular focus for example on privacy, others on usability and so on. Some of these Custom-ROMs are only executable on certain devices, others support a variety of devices.
On this page you find a collection of possible Custom-ROMs to choose from, each with a short description and a link to the original sources.
About the bootloader
Some phones also have a locked boot loader which prevents you from booting and installing other operating systems. If you want to buy an Android phone, make sure that the boot loader can be unlocked.
Note that on some devices a locked bootloader can be unlocked with clever hacks but their manufacturers tend to declare your warranty void when you unlock their device's boot loader. But, given European Directive 1999/44/CE, just the fact that you modified or changed the software of your device, is not a sufficient reason to void your statutory warranty. See Carlo Piana's and Matija Šuklje's analysis for more information.
There are also companies that ship their devices with an unlocked bootloader or provide you with the means to unlock it. These phones should be preferred for easy installs and usability.
List of Custom-ROMs
Replicant is a free software mobile operating system putting the emphasis on freedom and privacy/security. Since it does not use any non-free drivers, it currently works only a limited number of devices.
LineageOS is a very popular Custom-ROM that works on a huge number of different devices. This is because it uses proprietary drivers and firmware. Even though it is usually more free than the default software your Android device comes with, freedom is not its main objective. You do not need a Google account to use LineageOS, neither does it contain Google Play services. LineageOS works pretty good with the Free Software implementation microG. List of supported LineageOS devices
GrapheneOS "is a privacy and security focused mobile OS with Android app compatibility developed as a non-profit open source project. It's focused on the research and development of privacy and security technology including substantial improvements to sandboxing, exploit mitigations and the permission model." Graphene OS currently only supports Pixel phones. GrapheneOS does not support microG. List of supported GrapheneOS devices
CalyxOS describes itself as "Privacy by Design". It is not as restrictive as GrapheneOS though and optionaly allows the user to install microG. CalyxOS only works on devices with a verified boot, which means the boot-loader is locked and can't be manipulated. CalyxOS also offers to go back to your stock Android, if you change your mind in the future. You probably won't but it's good to know that you can. List of supported CalyxOS devices
/e/OS is "is a complete, fully “deGoogled”, mobile ecosystem". The idea is to form a privacy-enabled internal system for your smartphone with no Google apps or Google services accessing your personal data. Google services have been completely removed and replaced with microG. You can set up an account for advanced backup features. It does support a large number of devices and it is also possible to buy smartphones with /e/ pre-insalled. List of supported /e/ devices
If you want to add another Custom-Rom, please do, this is a wiki!