Amazon is opening up Alexa skill notifications to more developers as of today, though it is worth mentioning that developers who are interested in this opportunity to utilize the notifications for Alexa skills will need to apply to be a part of the program. Notifications for Alexa skills aren't completely new as there are some apps which already utilize the new functionality, such as Dominos which will alert you to the status of your incoming pizza order.
Amazon plans to roll out the Alexa skill notifications to most users starting in the first quarter of 2018, so this extension of supported developers aims to really crack down and help Amazon put the finishing touches on the feature before its public release next year. If developers are interested in applying to utilize the notifications for their skills there is only a short survey that has to be taken and Amazon says it would reveal those surveys and then reply back to the developer with the subset of skills that apply to the use case of the notification function and which meet the criteria and requirements.
If you're unclear on what those requirements are, Amazon seems to make it pretty clear what it won't tolerate for the use of notifications, and this pretty much breaks down to not using skill notification for promotional purposes or for advertising of any kind, and this is likely because Amazon feels like it would be an invasion of privacy for the user. That said Amazon does want the notifications for these skills to be personalized to the user and Amazon wants the notifications to be based on transactions like the status of a pizza order as described above. This means that any developers interested in using the notifications will want to consider whether or not their skill fits within that guideline, though Amazon does not state that the notification is required to be based on a transaction, just that it prefers it. These notifications will work for Amazon's line up of Echo devices such as the Echo, Echo Dot, Echo Show, and third-party Alexa-powered speakers. Amazon doesn't mention whether or not this would be compatible with the Alexa app that can be installed on smartphones.