In addition to traditional web parts still being supported (in either full-trust or sandboxed solutions) as Simon mentions, you can develop new app parts in an app for SharePoint 2013.
An app part is like a web part, and in fact is also made available to users in a host web's web part gallery. Users add app parts to pages the same way they would add web parts. One fundamental difference between an app part and a web part is that an app part uses an IFRAME to load the app part's content.
App parts, like web parts, can also have custom properties that users can set. More information about this, and app part development in general, may be found here.