Whilst it is perfectly possible to just run git locally you lose all the benefit of the "distributed" part of git which makes backup so easy but will also, most likely also give you a web interface for doing some admin functions and also managing permissions and security.
Probably the best known is GitHub, however GitLab is also a strong contender and has some distinct advantages over GitHub. Both GitHub and GitLab have free and commercial options but in the commercial space there is also Bitbucket which is an Atlassian product like Jira and Confluence.
There are of course many ways you can host yourself. Some hosting companies provide git options via a control panel, others will let you install it, or you can host it locally in your own network. For some testing work I was doing I used Git Server on Windows | GitStack as this gave me full control, did not put data in the cloud and let me complete my testing.