How to implement SSO in Salesforce1 mobile app?

For organization has implement SSO to login to Salesforce from web browser, organization can implement the same for Salesforce1 mobile app. There are two option for this:

1. Log in to a custom domain 
From log in screen, tap Log in to a custom domain

Enter your company 'My Domain' used for SSO.

App will prompt you to enter your company network (e.g. Active Directory) username and password.

You will be asked to allow permission access to Salesforce1 for iOS or Android

2. Add new Connection

From log in screen, tap gear icon at top right

Tap + icon on top right corner

Enter your SSO URL (e.g. mycompany.my.salesforce.com) into Host and your company name (preferred) into Label, then app will ask your company network username & password, in the next screen permission to allow login using Salesforce1, this is the same with option 1 above.

The difference between "Log in to a custom domain" with "Add Connection", with add connection, you just need to add once and and keep using it, while login with custom domain, need you to type "my domain" every time you login. It is your choice since both works, but prefer to add connection once and you do not need to enter my domain every time you log out from the app.

Reference: Single Sign-On for Desktop and Mobile Applications using SAML and OAuth