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Google Apps Script is a scripting language that can extend the behaviour and interaction of applications in the Google suite:
Drive, Sheets, Documents
Calendar
Sites
Blogger, tasks, fusion tables...
It's not primarily aimed at 'professional' coders and programmers, but is a way for ordinary users to get more out of the applications - a bit like macros in the Office applications.
A bound script can live in the file it is used in (eg a spreadhseet) but scripts can also exist as a stand-alone file in Drive. As scripts part of the Google framework, issues such as identity validation, permissions and script safety are easily taken care of - normally quite a complex task with traditional web applications.
You can also make use of University-written 'libraries' of scripts, making the job of developing scripts easier and more consistent.
Apps Script is very similar to JavaScript, and like JavaScript you can develop code by drawing on extensive libraries of examples - though if you want to, you can learn the language and work at developing your own.
What can you do with it?
The phrase is wearing a bit thin, but the limit really is your imagination. Small and large tasks can be automated, allowing you to do things like:
reading information from a line in a spreadsheet and turning it into a document and emailing a link to a user
creating new functions for calculations in sheets
turning information submitted in Forms into printable documents
controlling process workflow
This depends on your existing experience with coding, and your preferred method of learning. A useful starting point is the Further Help & Support slide in this presentation:
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Take a look at this presentation that includes an overview of Apps Script
Visit the Google Apps script support pages