For starters, the nearest star to earth (other than our sun) is Proxima Centauri. It is approximately 4.24 light years away. This means that it takes the light from Proxima Centauri 4.24 years to reach us. When we look at Proxima Centauri, we are seeing what that star looked like 4.24 years ago.
The disk of our galaxy, the Milky Way, is estimated to be at least 100,000 light-years across. If you were to use a telescope to observe a star located at the opposite end of the galaxy from us, the light that you would see coming from that star would have traveled upwards of 100,000 years or more to reach us.
The closest galaxy to our own Milky Way is the Sagittarius Dwarf Spheroidal Galaxy. A satellite galaxy of the Milky Way, it is estimated to be about 70,000 light-years from Earth, traveling in a polar orbit (i.e. an orbit passing through the galactic poles) at a distance of about 50,000 light-years from the core of the Milky Way:
[CENTER][IMG]http://annesastronomynews.com/wp-content/uploads/2012/02/The-Sagittarius-Dwarf-Elliptical.jpg[/IMG][/CENTER]
The farthest observable supernova is SN UDS10Wil. It is estimated to have exploded more than 10 billion years ago:
[CENTER][IMG]https://upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Record-breaking_supernova_in_the_CANDELS_Ultra_Deep_Survey.jpg/800px-Record-breaking_supernova_in_the_CANDELS_Ultra_Deep_Survey.jpg[/IMG][/CENTER]
The farthest galaxy observed from earth (so far) is EGSY8p7. It is estimated to be 13.2 billion light-years from Earth. This means that it takes light from EGSY8p7 13.2 billion years to travel through to reach us.
The universe must be an old place for light from such distant objects to reach us. If one takes the view that the universe was created 6,000 years ago, then it must necessarily have been created with this appearance of age.