Twitter allows you to post short 140 character updates to the world. It's like a mini blog or text message sent to the people who follow you.
Your tweets can also be searched. Unlike Facebook you can follow someone without the need for them to follow you back.
All your tweets show up on your profile. You can access that page from a URL so anyone, even those without an account, can read your tweets. For example: http://twitter.com/SevenSharpTVNZ. (Fun fact: Capitals don't matter.)
I wanna go go-karting!Seven Sharp (@SevenSharpTVNZ) April 4, 2013
Like an inbox, your timeline shows tweets from all the people/accounts you follow. They appear in chronological order. That means if you follow 500 people and don't check Twitter for a day you'll miss some updates. That's part of what draws people to check twitter religiously.
If you want to alert someone that you're talking about them or specifically want to draw their attention to the tweet you can add their twitter username after an @ symbol.
Eg. Hey I like your show @SevenSharpTVNZ
This @ directs the message into a tab on their profile so they can easily see all the messages others want them to see. This allows people to have conversations on Twitter. But remember they're public conversations.
All public tweets can be searched using keywords. For example you could search for the term "earthquake" and every tweet mentioning that word would appear.
To help people find tweets about a topic, hashtags were developed. Hashtags are created organically - you can hashtag anything. People try to use similar hashtags so those following the same thing can see what others are saying about it.
Hashtags can also be used ironically for fun.
Eg. "Man, Dad has some really funny jokes. #Not"
What do I share?
That's up to you. You link to articles you enjoyed reading and think others would enjoy, announce news or share observations about the world. You can even tell people what you had for lunch (but please don't).
Can I share photos or videos?
You can. Through Twitter (and some other services) you can easily upload content and add a link to that photo or video.
Who should I follow?
We're thrilled to announce the birth of our son, Marco James Carter, born 28th March, happy, healthy & handsome :) twitter.com/DanCarter/stat&Dan Carter (@DanCarter) March 30, 2013
You can follow groups and organisations for updates on events, fundraising, etc.
Want to know more?
Twitter has a useful section for you to learn more.