The serverless term is probably among the hottest terms in the software industry lately. It may be described as the architecture style that partially or fully abstracts the infrastructure needed to run our software. This abstraction is usually provided by various third-party service providers.
To put it in the context of web application development, let's think about Single Page Application (SPA). Nowadays, we can easily develop an entire SPA on top of a fully managed infrastructure, such as AWS. Such a SPA may be written in Angular, having client components served from the S3 bucket, managing users through the Amazon Cognito service, whilst using DynamoDB as an application data store. The managed infrastructure abstracts away any hosting or server dealings from us, allowing us to focus our efforts on the application alone. What we end up with is one form of serverless application, depending on how narrow our definition is.