When we talk about the universe, we are really talking about one of two things:
- The observable universe, which is everything we can possibly see.
- The Universe, which is everything that has ever existed, currently exists, and will exist.
The observable universe has its own center, usually the Earth. It is a spherical region of everything that we can see, essentially anything whose light has reached us. We usually refer to this when we say things like "there are $10^{86}$ atoms in the universe."
In reality, everyone has their own observable universe, and it can change depending on where you are. An exoplanet far away has its own observable universe, and can receive light from different places. Essentially, you are the center of your own observable universe.
I assume you're talking about the latter, though. The Universe (notice the capital "U") is all of space and time and its contents. Anything that has existed, will exist, and currently exists is part of it.
The Universe is thought to be infinitely large, so it can't have a center. The center of something is the point equidistance from the edges, but if something spans infinitely long, it would just keep going. It wouldn't have an edge, and thus it wouldn't have a center. You couldn't find the point equidistant from the edges if it just spans infinitely.
You might ask, "then where did the Big Bang start? Surely it must've been the center of the Universe, right?" Well, you can say the Big Bang happened everywhere. Before the Big Bang, matter filled most of the Universe's emptiness. It was essentially dense, and so got extremely hot to the point where no hadrons could form.
It is thought that this temperature caused space itself to expand. Essentially, more space was created in between all the matter, until everything was able to cool down. This is what we call the Big Bang. It didn't happen at a certain point, but rather it happened everywhere.
EDIT: I understand your confusion. Let me just clear some common misconceptions:
The Universe is not like a ball. Rather, you can think of it like a flat grid, and its "expansion" just means that the distances between objects on the grid are getting larger. In essence, more space is being created between the objects. That's what we mean by expansion — that objects are moving away from each other, since more space is being created between them.
Here's an easy analogy: imagine you are walking your dog. Suddenly, the ground begins expanding between you. You and your dog will separated and continue receding away from each other.
That's essentially happening everywhere: space is expanding between everything, so we are drifting away from other galaxies. The Universe is infinite, and we can constantly drift apart from other objects because space is being created in between us. Here's a GIF I made that might help you get it:
You can see how the galaxies drift apart as the space between them increases. And this happens everywhere in the Universe. So let's keep that in mind and clear up the Big Bang.
The Big Bang did not happen at a single point, nor did the Universe begin at a single point. The Universe is, and has always been, infinite. The Big Bang was just when the Universe's expansion really began — that is, when objects started drifting away from each other. The Universe was still infinite, but there was less space between the matter.
This density caused the Universe to get extremely hot and expand. So here, space itself was distorted and began to expand. More and more space was created between the matter, and still is now (although now it is mainly due to dark energy instead of heat).