Define uniform expansion?
Uniform Expansion, in physics (or more precisely, cosmology) refers to the expansion of our universe. According to this model, at large scales the universe is expanding at a uniform rate. This idea is a consequence of Hubble's Law.
Edwin Hubble, after meticulous observations, found that galaxies are moving away from each other. The rate of motion or velocity is directly proportional to the distance of the galaxies from each other and the constant of proportionality is the Hubble Constant. This combined with the Big Bang Theory of origin of universe, physicists arrived at a large scale picture of our universe; one in which the universe itself is expanding and at a uniform rate. This uniform expansion also means that there is no center of the universe, as the space itself is expanding. Think of two dots on a balloon. Once we start injecting air into the balloon, the dots will recede from each other and the center of balloon will keep on shifting as it expands. Something similar is happening to our universe, but at a very large scale.
Kindly note that at local scale there may be some exceptions to this uniform expansion (e.g., galaxies attracting each other because of gravity). This phenomena is observed at a very large scale.
Hope this helps.