How far can we see into space? A very long way indeed. But not only are we seeing objects that are far away we are also seeing them as they existed (past tense). Why? Because as much as we would like to think light is instantaneous it actually travels at a measured or finite speed.
How fast is it? Light travels at 186,282 miles per second or roughly 300 million meters per second. Pretty darn fast but when you start talking astronomical distances the speed of light becomes apparent. Once you know the distance to an object you can then calculate how long it takes light to reach it.
The light reflected from the Moon takes 1.255 seconds to reach us here on Earth. Light emitted from the Sun takes over 8 minutes to get to us. And so on. The most distant galaxies in the Hubble Ultra Deep Field image are 13 billion light-years away.