Definition: Non-attachment is a consciousness endowed with renunciation that lacks attachment to samsara and samsaric perfections.
Its function is to act as the basis for not engaging into negative actions.
Non-attachment in the context of giving up the happiness of this life and directing one’s attention to the happiness of the next life is the motivation of the practitioner of the small scope.
Non-attachment to the whole of cyclic existence, giving up the grasping for samsaric perfections from the depth of one’s heart, is the motivation of the practitioner of medium scope.
Some thoughts regarding attachment:
Attachment is a minion of ignorance. From the Debate between Wisdom and Ignorance,
Ignorance says to wisdom:
They who belong to my retinue, [38bcd]
Which is dominated, among others, by the three poisons,
Jealousy, miserliness, pretension, dishonesty, conceit,
The sixty-two corrupt views, 
Pride, laziness, non-conscientiousness and others,
Such as great desire, I send them out continuously.
Just to subdue them is hard, even without severance.
I am the essential person in the continuum, [40abc]
You, wisdom, are the adventitious one.
If someone goes then it is your kind.
Ven Choden Rinpoche commenting these lines:
Also, I am not alone. I have a very numerous entourage. The different mental factors that are dominated by true-grasping and the other main ones such as attachment and anger, I send out continually.
Because I send them out, it is even difficult to just subdue a little those that are like me, i.e. which do not contradict me but agree with me: Jealousy, miserliness, pretension, dishonesty, conceit, the sixty-two appalling views, pride, laziness, non-conscientiousness, and the like. There is also great desire, where whatever possessions one has, it is never enough and where one always wants more.
It is difficult to just subdue or lessen these a little, without even talking about being free from them from the root.
Further, I am the essential person in the continuum, and you, wisdom, are the adventitious one. Hence, if someone has to go, then it is your kind.
Did not Einstein say that to repeat the same action over and over again, each time with the hope of a different result, is the definition of insanity?
While it is hard to give up samsara altogether, we should definitely be able to see that: SAMSARA IS OVERRATED.
Most of us would like to alleviate the sufferings brought on by our attachments, but if possible without giving up our attachments. We need to realize there is no middle ground. Either we cling onto the object, or we do not.