Mobile first indexing is the basic term of new Google rankings criteria. It means that Google checks availability of mobile version of your site and if it is available then Google first includes it in its index. Due to first indexing it has become the basis of rankings for any website. This is because you see mobile version of your site as cached version. In your Google webmaster tools you can also see most of the traffic coming to your site is pointed as from Smartphone Googlebot.
Why the term mobile first is used?
It is called mobile first because Google index is not wholly dependent on mobile version. If your site does not have a mobile version then your site will be indexed as desktop version. But it will definitely harm your rankings as mobile friendly sites now get more exposure and better rankings even in desktop search engine.
Until now, Desktop version was indexed as primary version and was considered as base of ranking. Mobile version was considered as secondary or alternate version of your site and Google had to cache both versions separately and was showing mobile url (m.domain.com) to mobile users. But now they have changed it and all webmasters must take care of this change to ensure better search engine rankings.
In this regard best practice is to make responsive designs so you have not to make 2 versions of your site and Google consider your site as one for whole to give you better rankings not only on mobiles but also on desktops.