Like all subatomic particles, electrons don't seem to have a discrete location; when we try to measure their location, we get different results each time, as you said. However, these apparently-random results follow a predictable pattern, so we describe it using an equation predicting where it is most likely to be found (the Schrodinger wave equation). For electrons in some stable states, that area is roughly spherical, meaning that there is some point where it is most likely to be found, a fuzzy area near that point where it is fairly likely to be found, and a distance from that point past which it is very unlikely to be found. We can therefore model the electron as being a tiny discrete thing that whizzes around in a roughly spherical area, which is the "cloud" in this article.
All that "can be modeled as" stuff is unwieldy, so people say the electron "is" a spherical cloud when they mean that modeling it as a spherical cloud is useful for understanding the matter at hand. What this article is describing is an attempt to measure whether that spherical region is perfectly spherical. If it weren't - meaning, if the electron was slightly more likely to be found on one side of the sphere than the other, then it would have an electrical dipole moment, which is what they were trying to find.