In SFDC, you have a Contact List View which shows Contacts sorted by LastName first, then FirstName. Yet, when viewing the Contacts in the Salesforce1 app, it is showing the Contact sorted by FirstName first, then LastName.
Once completed, open up Contacts in the Salesforce1 app via the Navigation Menu. On the lower right hand corner you will then have the option to sort the Contact List via "Last Name" or "First Name", in ascending or descending order, by selecting on the icon.
By default, Contacts in Salesforce1 are sorted by FirstName, then LastName.
In order to have your Contacts sorted by "LastName", then "FirstName" in the Salesforce1 app, just like in SFDC, you would have to make the following changes to the Contact List View:
1. Click on "Contacts" tab
2. Select the appropriate Contact List View
3. Click on "Edit" next to the Contact List View
4. Scroll down to 'Step 3. Select Fields to Display'
5. Remove "Name" from the 'Selected Fields' by clicking on the arrow
6. Move "First Name" and "Last Name" from 'Available Fields' to 'Selected Fields'
7. Make sure to move "Last Name" to the TOP and then have "First Name" listed right after "Last Name".
8. Hit "Save"
Once selected, your Contact in SFDC would show Contact Names sorted by LastName first, then FirstName and also the same way in the Salesforce1 app.