In case of the Euro area, this recession signals the end of an expansion that has lasted 57 quarters (or more than 14 years). The previous recession was in 1993. The CEPR does not recognize the 2002/2003 period as a recession although they admit that it was period of particularly slow growth. In the case of the US, the last expansion lasted 73 months given that there was a recession in 2001 (March to November).
Both the CEPR and the NBER follow similar methodologies to define a recession, they look for periods of "significant decline of activity" and "activity" is measured by a set of economic indicators (not just GDP). This definition differs from what many refer to as a technical definition of recession, two consecutive quarters of negative growth. As an example, in 2001, the US did not have two consecutive quarters of negative growth as it alternated from positive to negative but the NBER recognizes this period as a recession.
The reason why their announcement take place months after a recession has started is that they wait for very strong evidence of the change in the phase of the cycle before they make their call. They are not in the business of forecasting business cycles.
More information on the CEPR dating can be found here. The NBER methodology and announcements are on their web site.