I recently built an API using the API Gateway and Lambda. The API was built the "old fashioned way". Zipping and uploading Node.js files was painful, and creating body mapping templates in the API Gateway was time consuming.
Serverless is essentially a deployment mechanism that abstracts the API Gateway, Lambda and other products from your favorite cloud provider. It's a great way to create web API methods and cron jobs without worrying about Lambda functions, Cloud Watch events, and configuring with the API Gateway. Apex is another tool that provides similar functionality.