It feels like I have been a seeker forever, but have not yet found the enlightenment I have been seeking. Is enlightenment something that happens, or is it just an understanding?
Try not to make so much out of the idea of enlightenment. Awakening is in scale, and to answer your question, it is instantaneous understanding. For instance, every time you can catch "you" about to defeat you, this glimpse into your own inner workings is a form of enlightenment. Why? Because what was formerly in the dark now has light. Instantly your suffering is diminished. As a person begins to prefer this inner light to the false sense of self that inner darkness produces, it follows (in the wake of this new understanding) that what is untrue must perish. This is the true beginning of dying to ourselves. So it follows: We awaken first in order to die, and as we die to ourselves, there is a rebirth. Ultimately, it could be said that this rebirth, this fully established Light in a man or a woman, is what it means to be "enlightened."
I have had moments of spiritual awakening. Yet these moments seem to fade all too quickly. When such moments arise, I have a new depth of clarity and silence. Are there any specific lessons to apply during these moments to help anchor myself there?
Nothing is more priceless than those moments when we suddenly are able to see that the world we took as being everything was only a small part of a greater something. Suddenly, we realize we indeed dwell within ourselves with- in a much larger world than we could have hoped existed. When these awakenings come, it seems natural to want to hang onto them, but consider the following: In a certain sense, that moment of Light came unannounced. Its presence awakened you to itself and all that it revealed. Now "you" want to be the one who possesses this Light and who can call upon it at will. This "you" is what is standing between you and the Light and, as difficult as it is to do, you must work at letting go of every wish you have to return to any moment once glimpsed.