The interaction between moving bubbles, vapor voids in liquid, can arguably represent the simplest dynamical system in continuum mechanics as only a liquid and its vapor phase are involved. Surprisingly, and perhaps because of the ephemeral nature of bubbles, there has been no direct measurement of the time-dependent force between colliding bubbles which probes the effects of surface deformations and hydrodynamic flow on length scales down to nanometers. Using ultrasonically generated microbubbles (approximately 100 microm size) that have been accurately positioned in an atomic force microscope, we have made direct measurements of the force between two bubbles in water under controlled collision conditions that are similar to Brownian particles in solution. The experimental results together with detailed modeling reveal the nature of hydrodynamic boundary conditions at the air/water interface, the importance of the coupling of hydrodynamic flow, attractive van der Waals-Lifshitz forces, and bubble deformation in determining the conditions and mechanisms that lead to bubble coalescence. The observed behavior differs from intuitions gained from previous studies conducted using rigid particles. These direct force measurements reveal no specific ion effects at high ionic strengths or any special role of thermal fluctuations in film thickness in triggering the onset of bubble coalescence.