Being aware of what comes to your mind is often called being "mindful". Increasing mindfulness increases awareness of what is happening in the mind. Various sorts of meditation lead to an increase in mindfulness.
One of the most useful notions for mindfulness practice I have come across is Eckhart Tolle's question:"What is the next thought to arise in my mind?" Sitting and paying attention to what exactly is happening in the mind, alert to see what comes up next, can feel odd. I was just aware of my ankle. Was that a thought or not? Should I say the next thought turned out to be about my ankle or was that too quick and unimportant to count as a real thought?
Trying this sort of thing can make a person quite aware of personal processing speed, the time that elapses between noticing the ankle and recognizing the thought is there and what it is about. Jack Kornfield encourages his students to label. Consciously and deliberately thinking "ankle" gives a small step away from the idea, enough distance to recognize it and also enough to distance the thought enough. A little distance increases the perspective and offers an opportunity to recognize the thought for what it is without getting lured into a review of past, present and future hopes, sensations and fears about the joint between foot and leg.
The mind can experience so many thoughts and feelings nearly all at once that it can be impossible to name them all before they slip into the next thought that comes. When the mind is moving fast, by the time the label is thought of, something new has arisen, maybe something related like "Where did I put my book on ankles?" but maybe something quite different like "I haven't had a good cupcake for a long time." Kornfield recommends labeling what comes up but not trying to consult a thesaurus for the exactly right label. A label can be quite general, such as "mixture" or "confused".
Mindfulness practice also highlights how much happens in the mind and the body all the time. We stay physically still during the practice to encourage better concentration on the experiences that come to the mind, rather than ignoring them or glossing over them in favor of today's news or the worry du jour.