How to change the Master page of your SharePoint site in Sharepoint 2013

“Master Page” option from site look and feel is missing

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1. Inside the current site's Site Settings, navigate to Site Collection Administration > Site Collection Features.
2. Activate SharePoint Server Publishing Infastructure if it's available... if it's not, you may be using SharePoint Foundation. 
3. Navigate to Site Actions > Manage Site Features, and activate the same feature (which should now be visible.
4. Return to Site Settings. Under Look and Feel, "Master page" should now be available.

(This is under the assumption the "Server Publishing Infastructure" feature can be enabled in your version of SharePoint.)

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Changing Master page of your SharePoint site is a Common Task.In SharePoint 2013 these are various ways to Change a Master Page of your Site. You can use the Design Manager,Themes (Change the Look) or the old usual Master Page link.

Lets look at the Options –

Option 1 –

“Master Page” link – If you have Publishing feature turned on, you can navigate to Site Settings and under “Look and Feel”  select  “Master Page” link. here you can set the master page of your Site.

Option 2 –

Other option is to use the new Design Manager for setting a Master Page. Lets look at the Steps –

1. Login to your SharePoint site and navigate to Site Settings.

2. Under Look and Feel click on “Design Manager”.

3. Design Manager is a new tool introduced in SharePoint 2013 to add your own design package (solution\wsp for custom master page) and then set it as your master page. well for here we will just discuss how to change a Master page to one that already exits.

In Design Manager click on step “7. Publish and Apply Design”.

Next click on “Assign master pages to your site based on device channel”

Now you get your popup to switch the master page.

Notice in System Master page it also tells you to select a Channel. A channel is for device specific page rendering or setting. You can create a channel for an ipad or surface from Site Settings > Device channels, which allows you to give the channel an Alias (for referencing in the UI and in code) and then “Device Inclusion Rules”. These rules are substrings that SharePoint will look for in the browser’s User Agent.

Option 3 –

As specified by Jonathan Kern (MSFT) you can also change the master page from Themes window. Simply navigate to Site settings then click on “Change the look”.

On the Themes page click on the theme that you are using.

Next on the left pane you can select the master page for this theme.

Next click on “Try it Out” and then “Yes, keep it”

and you’re done!