  Strong electron correlations lie at the origin of transformative phenomena
such as colossal magneto-resistance and high-temperature superconductivity.
Already near room temperature, doped copper oxide materials display remarkable
features such as a pseudo-gap and a "strange metal" phase with unusual
transport properties. The essence of this physics is believed to be captured by
the Fermi-Hubbard model of repulsively interacting, itinerant fermions on a
lattice. Here we report on the site-resolved observation of charge and spin
correlations in the two-dimensional (2D) Fermi-Hubbard model realized with
ultracold atoms. Antiferromagnetic spin correlations are maximal at
half-filling and weaken monotonically upon doping. Correlations between singly
charged sites are negative at large doping, revealing the Pauli and correlation
hole\textemdash a suppressed probability of finding two fermions near each
other. However, as the doping is reduced below a critical value, correlations
between such local magnetic moments become positive, signaling strong bunching
of doublons and holes. Excellent agreement with numerical linked-cluster
expansion (NLCE) and determinantal quantum Monte Carlo (DQMC) calculations is
found. Positive non-local moment correlations directly imply potential energy
fluctuations due to doublon-hole pairs, which should play an important role for
transport in the Fermi-Hubbard model.
