DPA attacks measure power levels at different parts of the chip and apply statistical analysis to overcome countermeasures, such as added noise, that are applied to obscure individual bits. Measuring the power usage identifies what kind of computational operations are being done by a device. An analysis will reveal several bits of the cryptokey at a time; the process is repeated to eventually produce the entire key.
DPA attacks are dangerous because they circumvent the hardware and software security that vendors have put in place. Because such attacks are non-invasive, it's possible for an intruder to compromise an embedded system without leaving a trace.
To avoid DPA attacks, experts recommend that enterprises use smart cards that will randomly generate a new key from the old key each time the card is used, thus rendering captured keys useless.