I will answer the second question first as it will help with the first.
The DG in 'DG flag' refers to the compass system Directional Gyro. This is a gyro (lets talk about conventional systems not laser gyros) which has its spinning axis in the horizontal plane and is fully articulated by means of a gimble system so that for any reasonable a/c attitude the gyro spin axis remains horizontal. The gyro direction is displayed on a compass indicator or HSI in the cockpit and in the event of a failure a flag will be displayed on the indicator too.
Now to the first question.
Any gyro is subject to errors caused by the earths rotation or movement over the earths surface. This is because the gyro rigidity (its ability to point in the same direction) acts on a point in free space and not on the earth. For a DG, the main errors are, apparent drift due to the rotation of the earth, or earth rate error. A tilting of the axis of rotation due to a combination of a/c movement over the earths surface and earth rotation. There are other errors too but they will also be corrected by the methods used to correct the main errors.
In order to prevent the gyro spin axis from being displaced from the horizontal, the gyro is made what is called 'earth tied'. For this, servo motors attached to the gimbal pivot points are fed with a signal from level sensors. Whenever the sensors detect that the axis has become displaced, a correction is fed to the relevant gimbal motor to restore it. In order to prevent false correction due to a/c acceleration or centrifugal forces, g switches are installed in the circuit to stop the correction when the a/c is accelerating or turning.
In order to maintain the correct heading indication (this is mainly error due to earth rate) another servo motor again on the relevant gimbal pivot point is fed with error signals but this time from a remote magnetic flux sensing device. On initial power up a fast slave system is used to quickly align the gyro. After this a normal slave rate of 2-3 degs per minute is used to maintain the gyro aligned to the magnetic heading.
In this way short term variations, due to the a/c turning etc are dealt with by the gyro and long term variations due to drift, earth rate, etc are dealt with by the magnetic sensing device.
Hope this helps.