This part is listed by BMW as BMW part number 11727557903 (11-72-7-557-903) and is described as Vemo Aftermarket Secondary Air Pump for N51 SULEV engine.
The N51 SULEV engine on 2007+ E82 128i and E90/E91/E92/E93 328i/xi models used an air pump to force fresh air through the control valve and into the exhaust, where it helps to reduce emissions.
If the pump fails, too much carbon-rich exhaust gas flows to the catalytic converters, causing premature failure of the cats and O2 sensors and triggering a fault code and check engine light. There are other components to the secondary air system but the pump is a fairly common replacement part.
This part is only for SULEV (Super Ultra Low Emissions) models, otherwise known as California spec.
BMW Fitments:
1 Series
E82 / E88 (08-13) > 128i
3 Series
E90 / E91 / E92 / E93 (06-13) > 328i
E90 / E91 / E92 / E93 (06-13) > 328xi