The organization that designates when our economy is or is not in a recession is something called the National Bureau of Economic Research. It is a private research organization that does research on important economic topics.
The NBER determines whether the US is in a recession by studying the real Gross Domestic Product on a quarterly (four times per year) basis. If real GDP declines for two straight quarters, the NBER declares that we are in a recession.
The reason that the NBER does this, rather than having the president or an organization that he (or someday she) controls is that it is important for the decision to be made by an objective observer. Imagine, for example, that an election is coming up. The president would be extremely interested in making sure that the economy was not officially in a recession during the election campaign. This would put a lot of pressure on the government agencies to come up with reports that were politically advantageous for the president.
Therefore, we have the NBER do the reporting. This ensures that reports about recessions will be made on an objective basis, not on the basis of political needs.