Sharepoint - How to activate "SharePoint Server Publishing Infrastructure " in SharePoint Online

you should find "SharePoint Server Publishing Infrastructure" under

"Site settings" > "Site Collection Administration" > "Site collection features"

you wont find it under site feature

or you can access it through this link :

https://{sharepoint server}/_layouts/15/ManageFeatures.aspx?Scope=Site

Things are a little different on SharePoint Online (SPO) than on SharePoint Server. You find features in the Site Settings page (equal to server version). But in the SPO version, the manage site feature link is located in the section Site Actions. You can browse to it directly via the URL

https://<site collection url>/_layouts/15/ManageFeatures.aspx

Unfortunately, the name is different on the SPO version which may cause confusion. Activate SharePoint Server Publishing on your site, and you'll have expected functionality of the SharePoint Server Publishing Infrastructure feature.

Go to Site Settings->Site Administration->Site Collection Features-> SharePoint Server publishing site infrastructure

Check this link: It will help you understand what all it does

http://www.sharepointchick.com/archive/2011/06/23/sharepoint-publishing-features-functionality.aspx