This is surprisingly one of the most frequently asked questions we get. Please take a look at this picture and look out for the blue New Thread button on the mid-left side of the screen:
You *MUST* be logged in to start a thread - this obviously enough means that you must have a Boards.ie account to start a thread. You'll know you're logged in in the second area on the upper right that I've highlighted with "your name here" in it.
You can only start a new thread when viewing a forum, not when you're reading a thread.
Finally, there is no "New Thread" button on our front page which seems to confuse a lot of people, but having one there makes no sense as the front page is not a forum, merely a listing of the most recent threads on the site across all forums.
Thought I'd update this as the process is different if you're using the touch site.
You still have to be logged in to post a thread, but see below screenshots for explanation.
How to start a thread on the mobile version of boards
You don't have to find a home for your post before writing your post, you can write first, find forum after if you like.