Because this is a standard salesforce object, we get to it by going to Setup / App Setup / Customize/ Contacts / Fields.

In Lightning: go to setup / object manager and select the Contact object, then click on "Fields & Relationships".  

Any new field we add to the record will be a Custom Field, so we click on the NEW Button in the "Contact Custom Fields & Relationships" section.  (In Lightning all fields, both standard and custom are rolled together on the list, so just click "New" to create a new field.

Depending on the field you are adding, there are a number of field types that can be used.  

(For help on field types, consult Salesforce Help.)

In this case, we want to add a picklist with the T-Shirt Sizes, so select picklist and click next.

Give the field a label name, and then enter the picklist values.

You can choose an alphabetical sort order, or set the first value to be the default.

Description is an internal field for you if the field needs explanation.

Help Text appears as a help pop-up next to the field itself.  Recommended that you use these for any field that isn't completely self-explanatory to make it easier for your users.

There are a LOT of standard Salesforce profiles that will appear in this list, so in can be easier to just make the object visible to ALL profiles.   But generally the two main profiles used by HOC users are System Administrator and Staff.... so selecting these two are sufficient.

Note: The list of profiles in your instance may vary.

The two key profiles to either grant or not grant access to are:

- System Administrator

- Staff

Exclude any of these you do NOT want to give access to.

You can also specify READ ONLY access if you want a given profile to SEE the value selected in the picklist, but be unable to change its value.

In this case - I've make Staff able to see the field, but NOT to edit it.  

Select the page layouts you wish to make this field available on.

Your list of page layouts may vary depending on which page layouts you have installed or created in your system.  You may be only using one page layout for all profiles -- or you may have a separate layout for System Admins and Staff.

By default, the new field will be placed as a new field in the top segment of the page layout.  To learn how to place the new field in a specific place of your choosing, see the post on Modifying Page Layouts.

After adding custom fields, they won't automatically show up in your existing reports.

To add custom fields in reports - you'll need to add the fields in the custom report types that pull the fields for the object.

See this post on how to control the fields that are available in report types.