Basically, an attachment is anything, any one, any situation, etc. that we don't want to lose or let go of. And this is an awful lot much of the time.
When we are attached to an outcome (a relationship or situation for example), we need to control things in order to try to get the income we think we want. We want to know how things will end up. Not knowing is unbearable.
When we are attached to things, anything physical (a material thing, person, a pet), we can't bear losing that thing, that person, that pet. We just don't know how we would survive life without it. We NEED it. Oh no, we may fall apart without that.
Letting go of attachments is a major change and we don't like change (as a species). We get comfortable and we don't want to lose that comfort. We identify ourselves by our attachments.
There are times that the universe prompts us to "let go" of a certain attachments. WE don't like that prompting, so we ignore it. Sometimes we "listen" to the prompting, and let go, and the door opens for us to have a new opportunity in its place, which can be an even greater "thing" than we had before. But, if we had not let go, we may never have gotten that new, wonderful thing. If we refuse, if we fight, if we put all of our energy into not losing that person/thing that we so desperately just cannot lose, and we lose it anyway, we are just so devastated. We feel destroyed.
UNTIL, we realize that those are just things. They are just situations. They are not us. Once we come to understand how the universe works, that those things happen for a reason, and fighting it only deepens our despair, then we can learn to let go of that control we so feel we need to have (because technically we have no control at all individually). The more control we fight for, the more we hopefully realize how little control we really had. Things go so much easier when we let go, we go with the flow of the energy of the universe. When we are in a place of perfection, total understanding & trust, when we have come to a place of truth, honesty, integrity, love... then we understand that everything happens exactly as it is supposed to happen. We turn over total control to the universe and we begin to "follow" the leads. We learn to trust in something that is higher, greater, bigger than us all. We come to the place of being one with the universal consciousness and totally getting "it". We no longer need to have control. We gain that by letting go, as we are universal consciousness.