Concentration's Meaning I've read a few answers and concluded that for most of these answers, concentration in meditation often implies unawareness of peripheral stimuli. This confuses me. I usually have wisps as thoughts rather than full blown thinking in meditation, but I don't particularly become unaware of my environment. 

Is such an experience very advanced? More so, does unawareness imply literally not seeing/hearing/etc. or merely not being solicited by such distractions?

Finally, I'm wondering whether the diminution of thoughts in itself means little. Is attention placed upon the object consistently more important than thought reduction? For example, I can imagine attention being close to the object but thoughts (e.g. about the object) arising, but I can also imagine a thoughtless person's attention being fragmented, directed without focus. Is any of these more important?

Thank you
Concentration is a poor translation of samadhi. Samadhi means meditative state. And right samadhi is one with strong mindful awareness. Deep samadhi is a state of absorption into the meditation object. 

There are also characteristic states of samadhi called jhanas which generally implies prominence of a particular mental quality (physical pleasure, mental joy, contentment and equanimity). However, the depth of each of these varies and are often debated between schools. This brings in the classification between commentarial jhanas (very deep states) vs sutta jhanas (just enough absorption to investigate phenomena).

Some schools prefer complete absorption as you imply in your question. Others, just enough absorption to have stable attention and plenty of awareness for insight practices. Certain suttas imply this sort of samadhi:


  Monks, secluded from sense pleasure... a monk enters and dwells in the first jhana. He steeps, drenches, fills and suffuses his body with the rapture and happiness born of seclusion, so that there is no part of his entire body that is not suffused with this rapture and happiness. Just as a skilled bath-attendant or his apprentice might strew bathing powder in a copper basin, sprinkle it again and again with water, and knead it together so that the mass of bathing soap would be pervaded, suffused, and saturated with moisture inside and out yet would not ooze moisture, so a monk steeps, drenches, fills and suffuses his body with the rapture and happiness born of seclusion, so that, there is no part of his entire body that is not suffused with this rapture and happiness born of seclusion. (D.i,74) 


There are two ways we typically know the presence of thoughts (The Mind Illuminated by John Yates PhD et. al). Attention and awareness. You attention can be stably on the breath and you mindful awareness can see thoughts and distraction come and go. This shows maturity of practice and there is a sense of standing aback and watching the breath. Thoughts stop arising as you consistently ignore them. It's a learned behaivour (unification of the mind aka citta-ekagatta) where the mind system is together working on stabilizing attention and observing the meditation object. This is the required amount of samadhi per a lot of teachers to begin insight training as your mind is now a sharp sword that can cut through anything without getting caught up.

Further and more in depth discussion on this topic can be found here: 


https://www.accesstoinsight.org/lib/authors/gunaratana/wheel351.html
http://www.leighb.com/jhanas.htm


The bottom line though is that samadhi is a joyful, restful, alert and mindful state. It should ideally be nourishing and nurturing.