Answer:
They're both imaginary, and they're parallel.
How to explain their relationship . . .
Imagine the earth's equator ... the circular line drawn around the earth's fat middle, exactly half-way
between the North and South Poles. Now imagine that the equator starts to get bigger. It's still lined up
with the same line on the earth, but its diameter is growing, and it loses contact with the surface, and keeps
growing, until it's over everybody's head. Now ships are sailing under it on their way north or south, and
it still keeps growing, like an enormous hula hoop (does anybody still know what those are?).
After a few hours, the equator gets so big that we can't really tell any more that it's only a few hundred miles
out there. Now it looks like it's a line on the solid surface of the sky ... the same surface that the sun, moon,
and stars are all painted on. Everything drawn on that surface looks like it's the same distance above the
earth ... the surface looks like the inside of a big globe. And it has that new line all the way around it, exactly
above the earth's equator at every point, but up in the sky. That's the celestial equator.
Another way to visualize the celestial equator ... maybe not a lot better than the first way, but here it is anyway:
Picture a gigantic knife, big enough to come along and cut the whole earth in half.
If it's big enough and comes in exactly right, it can cut the earth exactly on the equator, so you separate the bottom
half from the top half. Those are the north and south "hemispheres". Now, each half of the earth can sit flat on a table,
and the outside of the circle that it makes on the table is the line that used to be the equator.
OK ? Good.
Now if you will, picture an even BIGGER knife, one that makes the first one look like a boy scout's pocket knife.
This one is truly ginormous, almost too big to imagine. It can slice stars, solar systems, galaxies ! We're going to
slice the earth in half again with this one, and it has to be a clean cut. So we back way off almost to infinity, and
carefully line up our shot, so that we won't have to make any adjustments on the way in. When we're perfectly
lined up, we make our move. We keep our knife flat, come in smooth and steady from infinity, hit the equator
exactly, and slice precisely between the hemispheres. Then we follow through, and keep going off to infinity
on the other side, holding the knife flat all the way.
We have not only cut the earth in half exactly along the equator. This time we have also cut the whole
celestial sphere in half, on a line exactly parallel to the earth's equator. That line is (was) the celestial equator.