Our cells do biologically age as per normal regardless of our speed.  So if you are aboard a spaceship travelling away from Earth at 90% of the speed of light, you will see that you age one year for every year on your calendar.  Your hair will grow at the same rate you are used to, your nails will grow, your skin will keep growing and flaking off at the same rate as when you were on Earth.  Your internal bodyclock ticks away just as before.

The explanation given by 0urob0ross_ has one error. To the observer travelling in the space ship as he/she observes the earth speeding away, the time on earth will appear to pass SLOWLY (and not fast es explaiend by 0urob0ross_ ).

Please note this, in special relativity, there is no acceleration, and the observation of time dilation (and length contraction) is symmetric. If you remember the formula it contain the square of the velocity v (v^2), hence it doesn't matter if it is +v or -v.

If you return to earth, then youvhave to accelerate against the velocity and that causes the asymmetry and it is during the accelration that more time will pass on earth.

>> whether we travel at the speed of light

This phrase is meaningless. You cannot travel at the speed of light, or half the speed of light, or at any speed at all. You cannot stand still. These things are impossible because there is no fixed background to measure your speed against.

What you can do, is travel at a certain speed relative to something else.

And this answers your question. So long as you are not accelerating, you will not be aware of any effects. You will seem to age naturally by your own watch.

But if you compare yourself to someone on, say, a planet, when you are passing that planet at a high speed; then you will see them age more slowly than yourself. And they, looking at you, will see the opposite; you will seem to age more slowly than they.

Note that I put you moving past the planet, not towards or away, so that there would be no question of delayed light.

No speed does not affect your ageing cells

First of all remember that time is really just the measure of matter in motion. If you lived in a universe where no particles were moving with respect to each other, there would be no time – even if you were some sort of incorporeal being somehow observing all this. Time is simply the measure of particles of matter moving relative to each other. It is not a mystical fourth dimension that only starships enter.

Anyway, what basically happens with relativity is that the molecular motion of an object slows down as the object goes faster and faster. “Time” seems to slow down but what is really slowing down is just all the subatomic motions of all the particles of matter that make up the object that is traveling faster. And this slowing down is uniform within the object, so that the object itself senses no change.

For example if you were traveling in a spaceship at 0.5 times the speed of light, all the molecular motion in your body and within the space ship would slow down. Consequently your metabolic processes would slow down, the synapses in your brain would fire slower, the electrons moving through the ship’s computers would slow down, all the machinery in the ship would run relatively slower, etc., etc.

But since everything within the spaceship was slowing down at the same rate, within the confines of the spaceship you would notice no change. Even your conscious mind would notice no change since your own mental processes and physiological systems would slow down relative to everything else around you.

Once again “time” is just the measure of matter in motion. This is why cesium clocks run a little slower when orbiting around the earth in the space shuttle compared to an identical cesium clock left stationary on earth. There is nothing magic about this. The faster an object moves, the slower its molecular motion will be. A cesium clock measures the regular oscillations of a cesium atom. This clock is a precise enough instrument to measure the very slight relativistic effects of going at orbital speeds. Humans cannot perceive this difference.

But at near light speeds (if somehow people could travel them), the “slowing down” effect is substantial. The molecular motion of objects traveling at near light speeds slows down to a crawl – perhaps reaching a near standstill as one approaches light speed. Thus assuming somehow this could be done, some astronauts traveling at light speed to our nearest star (4 light years away) would experience only a few days or hours of time during that trip. Then maybe they would spend a normal year exploring the place. Then another few days or hours of time on the trip back to earth. During that time though, their NASA friends on earth would have aged 9+ years while the astronauts experienced only a little more than one year of time due to the exploring of Alpha Centauri.

There is nothing magic about this. It is just the way molecular processes work in objects that are in motion relative to a stationary observer.

Anyway that is my understanding of this phenomenon.