#chatweekly is a twitter Chat.

What is Twitter? 

Hear it from the horse's mouth: https://twitter.com/about

What is a Tweet? and how do you post it?

A tweet is a 140-character message posted via Twitter. Read on: How to post a tweet or also called as twitter update.   

What is a tweet chat?

A tweet chat consists of online conversations(chats). A specific #hashtag is generally attached to the tweet chat. There are tweet chats that happen at a specific day and time of the week or month regularly. Anyone can join tweet chats using the #hashtag specified. 

There is also a twitter application called tweetchat which you can use while participating or following these sessions. You can try Nurph  also.

Why twitter chat?

Follow the discussions.

Participate on the topics of your interest. Brainstorm. Learn. Listen. Unwind.

Connect with people. Get some visibility for yourself.