Relativity of Time Dilation

[gsingh2011]

So what I've heard so far is that if you're in a spaceship and you're moving at a speed close to the speed of light relative to the earth then what seems like 1 hour for you could be something like 1 million years for people on earth. This made sense to me until I realized you're moving relative to the earth, so if you saw a clock on the earth it would be moving slower for you. So why does one hour in the spaceship take 1 million years on earth and not one hour on earth take a million years on the spaceship? If they're relative to each other both make sense...

[Doc Al]

So what I've heard so far is that if you're in a spaceship and you're moving at a speed close to the speed of light relative to the earth then what seems like 1 hour for you could be something like 1 million years for people on earth.
OK.
This made sense to me until I realized you're moving relative to the earth, so if you saw a clock on the earth it would be moving slower for you.
Right!
So why does one hour in the spaceship take 1 million years on earth and not one hour on earth take a million years on the spaceship?
But it does! According to spaceship observers, earth clocks run slow; according to earth observers, spaceship clocks run slow.
If they're relative to each other both make sense...
Correct! The effect is completely symmetric.

(Note that to fully understand this, you'll need to consider length contraction and the relativity of simultaneity as well as time dilation. These three effects work together.)

[gsingh2011]

The reason this didn't make sense to me is because I was thinking of the twin paradox, and I was wondering why only one twin got older. If both clocks were moving slow relative to each other, then why does only one get older? Is this due to the length contractions and simultaneity parts that I am missing? I don't see how...

[Doc Al]

The reason this didn't make sense to me is because I was thinking of the twin paradox, and I was wondering why only one twin got older. If both clocks were moving slow relative to each other, then why does only one get older? Is this due to the length contractions and simultaneity parts that I am missing? I don't see how...
In the case of the twin paradox, the symmetry is broken: One of the twins must turn around and thus accelerate, while the stay-at-home twin never accelerates.

Please do a search on "twin paradox" and you'll find many detailed discussions.