When doing Weightlifting (Olympic Lifts, OL), there are many technique driven cues that we use. Triple extension is the big one; although I could argue that we really don't need to stress the ankle extension... that's another post! Keep the bar close, sit back, stay on your heels, explode, quick move, etc, etc. Getting under the bar is what I want to talk about, though. What does it mean?
First, we do a lot of front squatting (with a clean grip) for several reasons. 1) It is a great teaching tool. If you don't use DB Front Squat with beginners, give it a try. 2) The torso stays more upright. This helps to teach good back management and strengthen the hip girdle as it supports the core in the right way. 3) It helps you to get under the bar when doing a Clean. We also teach OH Squat for similar reasons.
If you can triple extend and get the bar to at least the chest (on "power" movements), the pulling work is done. Now you need to get under the bar. You will hear this described as "jumping down". I prefer "pull yourself under the bar". Once you have completed the pull, as described above, envision the bar suspended in the air. Rather than trying to pull it any higher with the arms (remember, you are already extended), pull yourself under the bar! Use the weight to accelerate down, pulling on the bar. This is a great cue for those that are having a problem dropping into their front squat (on Clean) or OH Squat (on Snatch).
I have gotten comments from athletes that doing combo lifts (Hang Clean & Front Squat or Hang Snatch & OH Squat) have also helped them get under it. With novices, I will have them do a Hang Clean, reset, then Front Squat. Once they have mastered this, I tell them to combine the two into one motion. Once they have that down, I'll start using the "pull under the bar" cue.
We do mostly Power moves. Power in Power Clean implies a high catch, ie not a deep squat. If you watch competitive lifters who do the full Clean (catching deep) they will start jumping down or pulling under the bar when the bar is about at the waist. This is a more advanced technique that many in college athletics will not achieve, but that is OK.
There are a lot of ways to skin this cat, but this one works well for me. Good Luck!