Drift alignment is a subject that causes a lot of confusion for amateur astronomers.  There are a plethora of articles on the subject available on the net some of which are confusing and some of which are well written and well presented.  

The first thing to clarify is what is drift alignment?

If we think about an astronomical equatorial mount, it has two axes that are perpendicular to each other.  One of these axes, the right ascension axis, is aligned so that it is parallel with the Earth's rotational axis.  By doing this and arranging a drive mechanism that matches the speed of the earth's rotation, a correctly aligned mount will track any object that the telescope has in view.  The other axis, the declination axis allows us to move the telescope to capture any object we wish to view.

This is fine in principle, but if the alignment isn't perfect then the object in view will drift in in the telescope's eyepiece until it moves out of view.  For normal observation, this doesn't present a huge problem as a small adjustment can be made manually to maintain the view of the object.  However, for astrophotography this is a big problem as the apparent movement of the object will cause trailing rather than a pinpoint sharp image.

Polar axis alignments can be misaligned in two ways.  The axis can be off in azimuth meaning that is is pointing W or E of the pole, or it can be off in altitude meaning that it is pointing N or S of the pole.  When we drift align, we correct each error one at a time until all error is eliminated.

Choice of star for drift alignment, or more importantly where the star is in the sky, is critical. Briefly, for azimuth alignment, a star close to or on the meridian and around +20 degrees declination should be chosen.  For altitude alignment, a star close to or on the equator and within 15 degrees of the eastern horizon.

Here's the method (Northern hemisphere):

Basic setup

Set up your mount and telescope as normal and align the RA axis with Polaris.  Get as close as you can as this will save time with drift alignment.

Fit an illuminated crosshair reticle eyepiece (Skywatcher have an inexpensive model that is suitable). Ideally you need a high power (200x).

Align the reticle so that your target star moves parallel to the crosshairs when the RA and Dec slow motion controls are used and Dec is up and down, RA is right and left.

Azimuth alignment

Find a suitable target star near or on the meridian and around +20 degrees Dec.

Get the target star in the centre of the crosshairs and allow the mount to track normally.  Watch for drift in Dec (up and down).

If the star drifts down, adjust the azimuth setting so that the star moves left in the eyepiece

If the star drifts up, adjust the azimuth setting so that the star moves right in the eyepiece.

Recentre the star and repeat the observation and adjustment until you see no drift for at least 5 minutes.  If the direction of drift changes to the opposite direction then you've made too much adjustment and you need to adjust in the opposite direction.

The faster the drift, the greater the error and the greater the adjustment required.  Practice and experience will guide you in the amount of adjustment required.

Altitude alignment

Find a suitable target star near or on the equator and within 15 degrees of the eastern horizon.

Get the target star in the centre of the crosshairs and allow the mount to track normally.  Watch for drift in RA (up and down) ignore any east-west drift.

If the star drifts up, adjust the altitude setting so that the star moves down in the eyepiece

If the star drifts down, adjust the altitude setting so that the star moves up in the eyepiece.

Recentre the star and repeat the observation and adjustment until you see no drift for at least 5 minutes.  If the direction of drift changes to the opposite direction then you've made too much adjustment and you need to adjust in the opposite direction.

The faster the drift, the greater the error and the greater the adjustment required.  Practice and experience will guide you in the amount of adjustment required.

Repeat the azimuth check and as long as there is no drift over for a least 5 minutes, the process is complete.