Throughout my development career I have used a lot of useful Javascript libraries for various reasons. Below I have compiled a short list of the ones I have found the easiest to use and most useful. They are in no particular order.

Datatables

This library has become quite well known and is a really good way of quickly building interactive HTML tables. Datatables is a JQuery plugin that essentially does what it says on the tin. It will create a table with headings and pagination given a JSON object of data, the JSON object can be on page or from an AJAX response. Due to its short learning curve, but full feature range I would say that is why many people are finding it a say useful Javascript library to have in their development stack.

datatables

Moment.js

Javascript dates and times are always a little fiddly, but Moment.js makes them really easy to work with. It has some great configuration options and simple to use functions. After only spending around 5-10 minutes reading through their documentation I had learnt enough to get started. Good documentation is a must in order to make a good Javascript library.

Moment.js

Full Calendar

Full Calendar is a calendar / events based Javascript library. It works a lot like Google calendar or if you are a mac user then the calendar app on Mac OS X. You can have month, week or day views, have events span across multiple days all with some very straight forward configuration. Again great documentation and really easy to get started with. Anyone making an admin interface or events section on a website would find this library really useful.

full calendar

AmCharts

This one is really powerful. AmCharts have a full selection of chart types from basic line and bar charts to complex multi series real time data feeds. They even have a map library as well. I once worked on an analytics platform and this was a must have Javascript library during that project. If you are comfortable working with JSON data in Javascript then this will be easy to pick up and a really useful tool to learn.

AmCharts

Parsley

Parsley is the easiest to use Javascript form validation I have come across. Load in the library, add a few data attributes and most of your needs will be met. They have a comprehensive documentation site if they aren't. One of the most commonly used Javascript libraries is JQuery validation, but I have to say Parsley is much better. It is easier to use and customize and has a just as good a following for when you need a StackOverflow post.

ParsleyJS