I know about the stock market and the foreign exchange currency market (forex) which is open 24 hours a day on weekdays.And then there is the bond market, and options, futures, forwards markets. What other types markets are there that you can trade in. Particularly low volume and low government control

Answer
All market will close in the weekend. One exception is gambling where you can also 'trade' your luck.

Is it really possible to make money trading on the FOREX market?

It seems too easy. You spend a few hours a week watching trends, buy, sell, and rake in huge money. It seems too good to be true. How many people are actually successful and how much time do they put in doing it?

Answer
Oh yes, it is of course possible to make nice profits from Forex if you know what you are doing.
$4 trillion a day worth of currency changes hand daily. So, there must be profits in this business.

I don't think few hours a week is enough. You need to read financial reports and track all the news coming about economy to do the trading.
For example, a couple of months ago, when the EURO was going down, almost every major currency (Except the USD, and the YEN which moves in the Opposite direction) was going down especially the Aussie and GBP.
But these days, this is not happending any more. The Euro is still at 1-year record low, while the Aussie and GBP are getting some push from their local economy.
There are a lot of reports that you need to read.

Better to start with books and follow some currency blogs.
Don't get cheated by some ADs that claim they can make huge money. If you do that, you will lose all of your money in a blink. Try to find some certified online Forex firms that charge you less money for the trades.