Notes Related List in Salesforce Lightning Experience

Activate 'Notes' in Lightning Experience

Classic: Go to Setup | Customize | Notes | Notes Settings | Enable Notes
Lightning: Click the Gear icon then go to Setup | Feature Settings | Sales | Notes Settings 

Select the "Enable Notes" option and click Save.

This is the only way to use Notes within Lightning.  If Notes are not enabled, the 'Notes and Attachments' related list in Lightning will only be available for Attachments.  

The Lightning Notes feature may also be called 'Enhanced Notes' in some Help documentation, which can lead to confusion. 'Notes' and 'Enhanced Notes' reference the same Feature.

Refer also to the Salesforce Help documentation on Notes.