dev.office.com/sharepoint is a great place to get started with the SharePoint Framework.
The SharePoint Framework is a modern, client side approach to SharePoint Development initially targeted at SharePoint Online in Office 365. Web parts created with the SharePoint Framework are a new type of web part and they can be made available to add on both existing SharePoint pages and new SharePoint pages.
There's a great hello world example for this process hosted at Build your first SharePoint client-side web part (Hello World part 1). All of the examples at dev.office.com are available for community contributions through github.
The basic steps of Hello World in the SharePoint Framework are:
Generate the skeleton of the project with the Yeoman SharePoint Generator.
yo @microsoft/SharePoint
Edit the generated code in the editor of your choice. Support for Visual Studio Code is strong across platforms.
Preview the web part using gulp and the local SharePoint Workbench
gulp serve
Preview in your SharePoint Online environment
Go to the following URL: 'https://your-sharepoint-site/_layouts/workbench.aspx'