HSDPA stands for 'High Speed Downlink Packet Access' and refers to a standard for downloading data quickly on a mobile phone.
HSDPA can offer theoretical download speeds of up to 14 Mbit/s and is often referred to as an 'enhanced 3G' service. It is deployed on networks that use the Universal Mobile Telecommunications System (UMTS).
A related technology to HSDPA is HSUPA - 'high speed uplink packet access' - which as the name suggests offers enhanced uploading transfer speeds from the phone to the network. The two technologies are sometimes referred together as HSPA - 'high speed packet access'.