What is 'relativity' in physics? Does it mean that a colonist living on the moon will age more slowly than her sister who remains on Earth?

We normally think of time as something constant: it passes by at a rate of one second per second, and nothing we can do will speed it up or slow it down. We can travel around in space, but not in time.

(Our subjective experience of time is different, of course: as Einstein said, ten seconds sitting on a hot stove can feel like an hour, an hour with a pretty girl can seem like ten seconds.)

It turns out, though, that time is not as consistent as it seems. Albert Einstein, about 100 years ago, first developed the theory of Special Relativity, and later the theory of General Relativity.

Special Relativity describes what happens close to the speed of light, and it turns out that when something travels very fast, close to the speed of light, mass increases, length decreases and time dilates. Time passes more slowly for the moving object than for stationary objects.

The trick, of course, is in the name: 'relativity' means all motion is relative. To the person moving, it feels as though they are stationary and the person who is stationary is moving so each feels as though the other should experience the time dilation. Resolving these apparent paradoxes is tricky, but it works... and we have observed time dilation, using very precise measurements of time from atomic clocks flying at high speeds in planes.

General Relativity deals with the way space-time - space and time together - behave in the presence of large quantities of matter-energy. 'Matter tells space how to curve, space tells matter how to move' is a way this is commonly described. Near a strong gravitational field, such as that produced by a star or black hole, space-time is curved, and time moves at a different rate. This leads to another form of time dilation. Time moves more slowly near a large mass, and more quickly away from it.

Your question says someone will age more slowly on the moon. I suspect they're thinking is that on the moon you are further from the Earth's mass and gravitational field and therefore time travels faster for you, but the earth is a relatively small mass so the effect is likely to be too tiny to notice.

I hope this answer helps you understand Relativity - both kinds - a bit better.