What is the importance of productivity in an economy?

To put it briefly, productivity determines how rich a
country can be.  Without high levels of productivity no country can get to the point
where it has a high level of GDP per capita and a high standard of living for its
people.

The reason for this lies in the definition of
productivity.  Productivity is the amount of value that a worker in a given economy can
produce for each unit of time worked.  In other words, higher productivity means that a
higher value of product is being produced by each worker each hour (or other amount of
time).

If you have low productivity, you need lots of
people to make amount X of money.  That money has to be distributed among many workers
and wages (and GDP per capita) are low.  The opposite applies in high productivity
economies.