A recession is a decline in economic activity that lasts long enough to be a significant event. One common definition of a recession is a decline in GDP that lasts for two or more consecutive quarters. Some economists argue that such definitions are arbitrary and that there is no qualitative difference between a recession and a depression (which is still more prolongued).
Recessions are often global as economies are now intertwined. A recession in one economy affects all the countries it buys from, by reducing demand for their products.
Governments can act to help recovery by increasing spending and cutting interest rates. What works and when is a matter of some controversy.
There is also an element of automatic stabilisation that comes from the rise in government expenditure relative to tax revenues when an economy shrinks.