What if the Twin Paradox use the "day time" and "night time" on earth as their age reference? I've searched this site, I found a similar question here but not exactly like mine.
So I can't understand the Twin Paradox when I use the "day" (bright time) and "night" (dark time) on earth as the age reference.
In the Twin Paradox it is said : 
the traveling twin is younger upon return than the Earth-bound twin.
Suppose the twin are 40 years old and can communicate instantly during the journey and the travelling twin calculate his age from his brother on earth standard. Say, when the travelling twin reach a star, the remain-on-earth twin told him that 7300 days and 7300 nights have passed on earth. (So the age of both of the twin are the same, 60 years old if in our common standard).
When the travelling twin return and almost landed on earth, the remain-on-earth twin told him that  (for example), 14600 days and 14600 nights have passed on earth. So they still have the same age, which is 80 years old if in our common standard.
So my question is:
What does it mean of "the travelling twin is younger" at the time they meet each other again -
while each of them think the same that his age is 29200 days and 29200 nights ? (80 years old if in our common standard).
Is the answer like below ?
Yes, their age is the same, but their appearance is different---> the travelling twin LOOK much much younger than his remain-on-earth brother.
 A: The paradox is that if the travelling twin takes a clock with him and consults that clock, instead of just letting the brother at home tell him what time it is, that clock will record less time having passed than what the brother on Earth tells him has passed.
It doesn't matter what kind of clock is used. It could be mechanical, a quartz crystal oscillator, a water clock, or whatever. He could even just count how many times he has to shave his beard. Because the problem isn't that the clock behaves differently on the moving ship. What's happening is that less time actually passes for the brother (and everything else) on the ship compared to on Earth.
Edit

The thing which I don't understand is : "a clock". Because to me, "a clock" is something like this : ...

This isn't how physicists define a clock. Currently we use the frequency of oscillation of the cesium atom as the reference to define the second. That means you build an oscillator locked to a particular absorption line of a sample of cesium gas, and when it oscillates 9,192,631,770 times, that's one second.
But it could work with any kind of clock. Say the two brothers build or buy identical digital wristwatches, that keep time based on a quartz crystal oscillator. Each time the quartz crystal oscillates 32,768 times, one second is recorded.
After the travelling brother returns, the earth-bound brother's wristwatch has counted off 315,360,000 seconds (10 years). The wristwatch that travelled on the space ship might have counted off 320 million seconds, or 600 million seconds, or 315 billion seconds, depending how fast the space ship travelled.
The reason for this is not because wristwatches malfunction when they travel on space ships. It is because more time passed for the space ship and everything in it.
It would work the same if they used mechanical spring clocks, cesium cell atomic clocks, pendulum-driven grandfather clocks (assuming the spaceship is designed to produce exactly 1 g of artificial gravity), etc. Even if the brothers just count how many times their hearts beat (assuming they do similar amounts of exercise, etc., to keep their hearts beating at a uniform rate).
The Earth-Sun system is just one choice of a system with a periodic behavior that can be used to indicate time. If you (some kind of super-powered alien) built a second Earth-Sun system and put it in a ginormous space ship and sent it on the same path as the brother's space ship, when it got back the Earth on the space ship would have revolved more times around the sun than the Earth that was left behind and didn't travel.
A: It happens because in spacetime different paths between two events can have different lengths through time.
It is analogous to different lengths of paths between two points in space. If you drive around three edges of a square and I drive along the fourth edge, although we start and end at the same point, our odometers will show that we have covered different distances.
Likewise, the travelling twin takes a route through time that is shorter than the route taken by the stay-at-home twin, so when they meet, their clocks show that they have travelled different distances through time.
A: Yes, I think you have it right. If you define the age of a person as the amount of time that has passed on the Earth since they were born, then the twins will of course always be the same age, because they were born on the same day. However the returning twin will have experienced less time passing, and will therefore be younger than his brother physically, mentally and in basically every other sense of the word, as Marco Ocram and The Photon have already explained very well.
