This is a complex question with an even more complex answer. But essentially there is no center of the universe according to current observations. If a center point were to exist, it would need to significant and special in a way that meant it was special to the whole universe collectively. So, this answer will contemplate the different ways that a center of the universe could potentially be created.
The most obvious first point of reference is that when an object is rotating you can pinpoint a center of rotation. This is the one spot on the object that stays still when the rest of the object is rotating.
If you picture this concept on earth, our center of rotation is along the axis, the line that connects the North and South poles. This same concept can be transferred to a spinning plate in a circus. The center of rotation would be the very middle of the plate, where the spike is interacting with it.
Similarly, a basketball spinning on a finger will also have an axis similar to the one that the earth spins on. The center of rotation would be where the ball is touching your finger. But with all this said, a sound way to detect a center of rotation, there are no observations to show that the universe is spinning as a whole, and so there is no way to define a center of rotation for the whole universe.
There is also such thing as a center of charge. This is a point on any object that exhibits an electric charge that is the same on average as all of the points in directions surrounding it.
This is easy to imagine on a sphere since the center of charge would simply be the center of the sphere. Sadly, the charge distribution in the universe as a whole is uniform on average. This means that there is no center of charge. And therefore, no center of the universe can be found this way.
The most obvious choice would be the center of mass of an object. Similar to the center of charge, the center of mass on an object would simply be the area which has an equal mass on average surrounding it. This is strictly for finite object though, that is objects that have a clear start and end.
But it is more confusing when it comes to infinite objects. When an object is infinite and uniform across the board when it comes to mass, you cannot define a center of mass since all points of mass measurement would be identical. But if you have an object that is infinite and not uniform, you can pinpoint the center of mass as being the center of the non-uniform part. To visualize this, imagine a length of rope.
On its own this would be uniform in mass. Now put a knot in it, and you have a rope with non-uniformity. The center of mass for the whole length of the rope can be defined as the center of mass for the knot.
An even more abstract idea for the center of mass concept is the idea of a cloud. You may think that it is impossible to find the center of mass in a cloud because they are so diffused, and the edges are not always clear. But you can define the center of mass in a cloud through measuring its density in particular parts of the cloud.
However, since observations show that the universe is infinite in size, it may have non-uniformities in the shape of planets and stars, but these are so far dispersed from one another that the universe actually appears more uniform. And by our own definition, and the definitions that scientists have put forward, an infinite and uniform object cannot have a center of mass.
On a more abstract note, an object can have a center of curvature. Imagine your mixing bowl for baking. It is curved but still has a center point. This center point is where the rest of the bowl curves away from.
You could roll a marble from any point on the rim and it would eventually fall still at the center point of the mixing bowl’s curvature, thus finding the center of curvature. But this concept is pulled up short by the fact that observations show that the universe is in fact flat and not curved.
The biggest possibility is to find the center of expansion. Since scientists have in recent years found that the universe is expanding, it follows that there could be a center of expansion. If you were to take a long reem of elastic fabric in the shape of a square and have someone weigh down any point by sitting on it.
You could then have four or five people pick up the edges and stretch the fabric. The center of expansion would be the area that the person is sitting down on. This is because this is the point of space that all other points of the fabric are moving away from.
Since we know that the universe is expanding thanks to multiple findings, these observations are also precisely what informed the idea of the Big Bang.
The theory is that: if the universe is expanding, there has to be a point back in time where everything began at one point, this could potentially be the center of expansion.
Sadly, even though this idea is gathering steam, this is not the case for the universe. The universe is expanding in every direction equally. This means that all galaxies are getting uniformly distant from one another.
This means that there is not a single center by which all of these things are expanding away from, rather that all things in space are staying the same but it is the space between them that is expanding.
Another theory could be that the area of the Big Bang is the center of expansion and therefore the center of the universe, but the Big Bang is the whole universe and nowhere at all. That’s hard to wrap your brain around but since the Big Bang started everything, it simply is everything.
The simplest way to define a center of the universe would be to pinpoint a specific object that exists in one spot alone. This could be a supermassive black hole or a nebula: a cloud of dust and gas in space that is visible in the night sky. Sadly, this theory also loses legs since objects like this are scattered throughout the whole of the universe.
The human race has the bad quality of thinking that we are the center of the universe, but all of these theories have debunked that idea for good.
The real answer, as unsatisfying as it is, is that the universe simply has no measurable center to it. Not one that can be measured by mass, charge or even expansion. This is because the universe is technically uniform, since the planets and objects that could make it non-uniform are so few and far between that they do not actually contribute to any sort of non-uniformity.