Density altitude is barometric altitude corrected for nonstandard temperature or pressure. So the Pilots Handbook of Aeronautical Knowledge says but what exactly does that mean?
First remember that an aircraft altimeter is really nothing more than a very sensitive barometer; it simply translates atmospheric pressure readings into those found at a corresponding altitude above seal level. Since air pressure drops by approx 1 inHg for every 1000 feet above MSL, a barometric altimeter becomes an effective means to gauge height above sea level or similar reference datum.
Since the atmosphere varies in both temperature and pressure in different places of the atmosphere around the globe, there exists a need for a standard atmospheric condition to be used as a datum of reference. In SI, this Standard Atmosphere is defined as 1013 millibars (29.92"Hg) at 15°C (59°F) at Mean Sea Level. This standard atmosphere is also a good baseline to calculate performance data for an aircraft at. Charts and performance data are usually created based on an aircraft operating in Standard Atmospheric conditions.
But since ambient atmospheric conditions vary in both temperature and pressure, it is convienent to have a means to convert the ambient air conditions and altitude into an equivalent pressure altitude in standard atmospheric conditions.
Hence this is what Density Altitude is.
Note that Density Altitude has nothing to do with your true altitude or absolute altitude; rather it's an equivalent altitude reading which one would see at STP in the current actual altitude, temperature, and pressure of the air you are currently in.
As an example, suppose you are at Lake Co. Airport in Leadville, Colorado. The field atmosphere here is 9934 ft ASL at 20°C with an altimeter setting of 30.12"Hg. These condition correspond to one you would encounter in air at standard atmospheric conditions at 12,614 ft ASL or gives us a density altitude of 12,614 ft. This doesn't change the fact that you are still only 9934 ft ASL while on the ground in Lake Co. but it does give us a means to quickly use the manufacturer's performance data based on STP to determine the aircraft's performance under these conditions. One would use said charts and compute the performed AS IF the airplane were actually at 12,614 ft at STP versus at 9934 ft in local atmospheric conditions.
Density altitude is of greater importance on hot days than cold ones because the air is less dense when warm than when it is cold. Since both the lift wings can produce at a given true air speed and the power output of an engine are directly proportional to the density of the air, this results in an aircraft which sluggish, even dangerously deficient performance on hot days with high density altitudes than on cold ones with low density altitudes.