Another (for an old universe):
The fact that distant starlight can be seen from Earth has always been a major problem for the young Earth idea. Because the speed of light is finite, what you are actually seeing when you look at an object is an image of that object from the past. "From the past" here has a few caveats regarding the relativity of our concept of the past, the future, and now. In the BBC Horizonprogram What Time Is It? physicist and former pop-synth player Brian Cox suggested that, as information cannot travel faster than light, and that time and space are relative, it can be considered that that the stars actually are what they look like "now", in a manner of speaking. Either way, though, the bottom line is still the same; the light has travelled a certain distance, for a certain time, before arriving on Earth to be seen by our eyes or telescopes. We can use this data to put a minimum time on the existence of the universe, by looking at how long some light has been travelling for.
On Earth, the delay caused by the speed of light is incredibly minor — when you look at an object a mile away, the light has been travelling for five microseconds. When you look at the Sun, you are seeing light that has been in transit for 8.3 minutes. It's more noticeable with sound and distant objects, but only because the light from things such as distant explosions or jet fighters is so much faster. There's still a delay and transit time for the information that says whatever made the light/sound must have been around that long ago to produce it.
On the cosmic scale of things, this delay is far from minor and really is noticeable. When astronomers look at the closest star to Earth (Alpha Centauri), which is roughly four light years away, they are seeing the star as it was four years ago from our perspective. When astronomers look at objects in the region of space known as the "Hubble ultra deep field", they are seeing the stars there as they were over ten billion years ago. Light we are receiving from these fields has been travelling for ten billion years, and the universe must have, therefore, existed long enough for that transit time to take place.
Therein lies the problem for young Earth creationism; if the universe is only 6,000 years old, how can objects billions of light years away — and therefore billions of years old — be seen?