Drag is unnatural movement of the fly by the tension of the line, whether it's caused by the current, the fisherman, or both. A drag-free drift is when the fly acts like it's not connected to a line in any way.
Overall, it's probably the toughest obstacle to fly fishing, not just for beginners, but the best of the best are merely a little better at it, but it's still a problem. And it's one reason streamers/buggers are so popular with beginners. Baitfish swim, hence, by pulling them in the water you are intentionally adding drag.
But with dry flies and nymphs and other stuff "in the drift", generally you want drag-free. There are 2 real enemies to this game.
1. Cross currents. This is when you're fishing across stream. Current speed varies between you and the fly, quite often you want the fly in slower water that's beyond the faster water. Faster water grabs the line, and pulls the fly.
2. Lip currents. One way to try and avoid drag is to cast upstream. But lots of times that means standing in the tail or the riffle below, and casting to the slower middle of the pool. Water speeds up as it approaches the tail, pulling the line faster than the fly should move.
Couple of tips.
1. For cross currents, you will always eventually get drag, but you can increase the amount of drag free time per cast by letting slack in the leader during the cast. There are many ways to do this. For a beginner, the easiest is probably to aim for a spot 3 or 4 feet ABOVE, as in straight up, from the target. Leader straightens in mid air, then just hold the rod, and it'll recoil a little before hitting water. As it recoils, that's the leader collapsing, and just let it fall to the water with s-curves in the leader. It'll take the current a few seconds to straighten out those curves, and that's your drag free time.
2. You should also learn to mend. This is when you get that C curve between you and the fly, pick the line up and throw it back upstream of the fly, trying not to do it so hard as to move the fly. When the line is in faster water but upstream of the fly, you have a drag free drift for a while.
3. There are a variety of more advanced casts. Aerial mends, curve casts, slack leader casts, parachute casts, etc. When you're ready for em, ask, but I wouldn't recommend attacking these just yet. Get the basics down.
4. Leader construction is extremely important.
5. Longer tippet helps with drag some. Also, if it's "micro-drag" (effect of current on leader, rather than fly line), finer tippets help too.
6. For most of these, there's a trade-off between avoiding drag and accuracy. Meaning, the casts and leader constructions which help you avoid drag also hurt accuracy. Which is more important is situational, hence the reason for so many different casts and leaders.
7. A great trick is an anchor. If rocks or logs stick out of the water, cast over them and lay the fly line on them. You've effectively shortened your line, all that matters is any drag between anchor and fly, which is closer to the fly and hence less change in currents. On small streams on a curve in the stream, often there's rocky banks. Use em to avoid lip currents, don't cast up the water, cast over the bank into the pool!
Sorry for writing a book! I'm done now.
