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In normal cosmology, 'comoving' distance or 'proper distance' is one of several distances measured by cosmologists to define distances between objects.
To talk about the shape of the Universe, scientists want to forget that the Universe is expanding. So they separate the shape from time and pretend that the Universe is not expanding. They remember that, really, the Universe is expanding, but they write this in a variable called the scale factor. Then, the distances they talk about, while putting the scale factor to the side, are called comoving distances. When they think about the Universe using comoving distance, they think of the Universe as not expanding. This way, some calculations are easier.
When the part of the Universe which we can see today started, it was very small. Everything in this part of the Universe was much closer together than today. It was also very hot and photons (particles of light) were moving in different directions. Light moves very fast, but the light speed is limited and the Universe was expanding very fast.
Think of two ants on the surface of an expanding balloon. One ant starts moving towards another, but the balloon itself is also expanding. After a long time the balloon starts expanding more slowly. Eventually one ant reaches another. That's like photons in our universe. Some of the light from the stars you see at night traveled for billions of years to reach your eye. Surprisingly, it started very close to where you were (or the atoms of which your body consists), but just could not keep up with the universe expansion. Of course, if something else started out very close to where you were, but was not traveling toward you, it would be very far away from you now because of the expansion.