The area you will need to click to add fields into a form will depend on which version of the CRM you are using. If you are using a version lower than V9 then you will need to click Settings > Customizations.
If you are using V9 and above, you will first need to click the Settings wheel at the top of the page and select Advanced Settings.
Once located, both Versions will be able to click into Customizations.
To be able to make these changes, your user will need to have permissions to customize your CRM (the out-of-the-box System Administrator security role contains the necessary permissions).
Once you have access, select Customize the System, then expand Entities
Find the correct entity that you would like to edit i.e. Contact and open the Forms section.
From here you can select the form you would like to add your field to, if you are using an older version of CRM (lower than V9) you will need to click more actions and select the edit option as seen below. Versions 9 and above will be able to select the name of the form they wish to edit.
Once the relevant option has been clicked, this will open the form editor. On the right hand side of the form editor, a list of all available fields will appear. Simply drag and drop your required field to the appropriate place on the form.
Once you are happy, save and publish your customizations.