Economic numbers: While economic numbers can certainly reflect economic policy, some reports and numbers take on a talisman-like effect: the number itself becomes important to market psychology and may have an immediate impact on short-term market moves. "What to watch" can change over time. In recent years, for example, money supply, employment, trade balance figures and inflation numbers have all taken turns in the spotlight.
These cover the bulk of countries outside Europe. Forex brokers catering for India, Hong Kong, Qatar etc are likely to have regulation in one of the above, rather than every country they support. Some brands are regulated across the globe (one is even regulated in 5 continents). Some bodies issue licenses, and others have a register of legal firms.
The foreign exchange market is where currencies are traded. Currencies are important to most people around the world, whether they realize it or not, because currencies need to be exchanged in order to conduct foreign trade and business. If you are living in the U.S. and want to buy cheese from France, either you or the company that you buy the cheese from has to pay the French for the cheese in euros (EUR). This means that the U.S. importer would have to exchange the equivalent value of U.S. dollars (USD) into euros. The same goes for traveling. A French tourist in Egypt can't pay in euros to see the pyramids because it's not the locally accepted currency. As such, the tourist has to exchange the euros for the local currency, in this case the Egyptian pound, at the current exchange rate.