Synapse is the most widely installed Matrix homeserver implementation. It is written in Python 3.
Instructions for installing Synapse can be found in the Synapse documentation.
Many guides have also been written explaining how to install Synapse, but there is sometimes confusion about the exact steps and it's best to read the official documentation above in case of any doubt.
Slavi has created a very useful set of Ansible playbooks which install containerised instances of multiple Matrix-ecosystem packages, the Synapse container being the centerpiece of these.
You can find the playbooks at https://github.com/spantaleev/matrix-docker-ansible-deploy, and there are instructions included in the repo.
To talk about the playbooks or get installation support, talk in #matrix-docker-ansible-deploy:devture.com.
It's also possible to use Docker containers without Ansible, just follow the instructions in the Synapse Docker README.
For further help join #synapse:matrix.org.