Short term growth is, as the name suggests, growth in the output of a country in terms of GDP over a given (short, usually a year) period of time. It is measured by the annual percentage change in GDP.
Long term growth however is when the country's productive potential is increased, the potential of the country's GDP is increased. Due to an expansion in eitherthe quality or quantity of factor inputs, the country is now able to produce more.
Both these concepts can be shown simply on an aggregate supply/aggregatedemand curve. SHort term growth would be shown by any movement along the x-axis (real GDP), and Long term growth shown by a shift to the right of the LRAS (long-run aggregate supply) curve.