There are plenty of creatures out there that have a sense of time. You only have to look at the seasonal migration patterns of birds such as the Swallow to realise that. But we humans are unique in the way that we use time. We have daily schedules, long term plans, a fascination with history, and a sense of eternity.
He has made everything beautiful in its time. Also, he has put eternity into man's heart’
Ecclesiastes 3 v 11
Our sense of time relies heavily on our ability to use language. To that extent the Piraha tribe from Brazil are very interesting. Their language has no numbers! It also doesn’t have a perfect tense (e.g. I have gone...) The way their language is structured thus translates into their culture. They only have a very shallow knowledge of their own history (no more than two generations) and there are no creation myths or similar such legends. As a people they live very much in the present.
So our sense of eternity is in part due to our language abilities. It’s also got a lot to do with maturity. A recent study confirmed what everyone probably knew already and that is that it isn't until we get to our late teens that we really start thinking outside of the moment and realise the long-term implications of our actions. Compared to adults, adolescents are less future orientated and less likely to plan for the future. What's fascinating is that the areas of the brain associated with foresight and planning continue developing through into the mid-20s.
From a faith point of view our far-sighted brains and erudite language combine to allow us to begin to comprehend eternity. The Bible says that eternity is why we're here in the first place!