How can I know that I am clear of thoughts while I am meditating? While meditating, one might try to be aware of their surroundings, to be present in the moment. But how can we become aware of it without thinking that we are aware? 
Another example is when one can listen to sounds or silence, and even if one is not trying to put stories behind the sounds, one might find oneself thinking about the wave of the sound, or its variations, its length, etc.
There are multiple kinds of mental phenomena, not all of which can be classified as thoughts.

First and the crudest kind, are visualizations, cravings, and replays. This is when we mentally leave here and now, and are experiencing what basically is a (short or long) daydream about the subject our mind is concerned or agitated about.

The second kind are mental chatter. This is when we think in words, silently saying them in our head. If you look carefully you will notice how the muscles of the tongue and lips are slightly engaged as you "pronounce" your inner speeches.

The third kind is conceptual thinking. This is when we mentally follow a chain of logical relationships between objects or concepts. These are still thoughts.

The fourth kind is labeling. This is when we mentally recognize phenomena we experience and identify them with a category or attribute that makes them distinct, given the context.

The fifth kind is registering. This is when we make a mental note of experience without labeling it to any category. For the purposes of this answer, these are the borderline between thoughts and non-conceptual experiences.

The sixth kind is non-conceptual perception. This is when we experience (see, hear, smell) or feel (a bodily sensation or emotion), without labeling, acknowledging or noting it in any way. Famous technique used by Zen masters to awaken one to this level is to make a sudden noise in the middle of long meditation session.

The seventh kind is perception of energy. This is when we feel chakras and meridians.

The eighth kind is samadhi. This can't be described in words and must be experienced to be understood.

When we say "my thinking has stopped", we usually mean levels 1-4, while levels 6-8 can still function. This said, it is not like in meditation our goal is to stop thinking. Rather, it is to open ourselves to levels 6-8. As we get better at this, thinking can go on "in the background" without affecting our meditation.