Bitbucket is a service for hosting version control repositories. It supports Git and Mercurial. Murcurial support will be phased out. It is a solution offered by Atlassian. It could be compared to GitHub, GitLab, or Bonobo Git Server.
This is Bitbucket’s cloud service. It gives you everything you would expect but in the cloud. It is hosted by Atlassian themselves which can be convienient but also means you give up some control. This would be a great place to start if you don’t want to make an upfront commitment in terms of servers and other overhead. It is also faster than setting up your own service. You can be up and running much more quickly.
Bitbucket Server was previously called Stash. It is a server that can be run on premise. This gives you everything you would get with Bitbucket Cloud except that you can run it on your own hardware. This is important if you need to keep all of your data onsite. This is a very real requirement for many organizations, especially those with regulator restrictions. This is often the case for anyone in financial services. Beyond this it is also nice to have more control over your repositories. Running your own server gives you this.
Bitbucket Data Center
This supports more enterprise features than Bitbucket Server.
Bitbucket vs GitHub