Sharepoint - Microsoft Teams integration with SharePoint Server 2013 Enterprise

For OnPremise? No. But if you have a Hybrid environment, it's doable.

It depends on the level of integration you're after. Currently we can add document libraries of any SharePoint Online site which will appear as a tab in the Teams global navigation, and where you have access to that document library and its files from Teams.

If this is the level of integration your after, you can add on premise libraries as well in your SharePoint 2013 Hybrid environment, just add the document library URL, and you're good to go

It is not available on premises. It's basically a chat-centered workspace in Office 365. It's not available as a standalone on prem product.

However, you can install the client application on your machine or phone. After you install and open it, it will ask for your work account or Microsoft account.

So in case of OnPrem users, you won't have work account. However, you can use the personal (Microsoft) account. So any user with the appropriate cloud license and who is in AAD (Azure Active directory) can login to Microsoft Teams, see other users' presence, send messages within Microsoft Teams, etc.

Reference - Introducing Microsoft Teams—the chat-based workspace in Office 365

Download link - Microsoft Teams download