The coalescence, and subsequent satellite formation, of two inviscid droplets is studied numerically. The initial drops are taken to be of equal and different sizes, and simulations have been carried out with and without the presence of an electrical field. The main computational challenge is the tracking of a free surface that changes topology. Coupling level set and boundary integral methods with an embedded potential flow model, we seamlessly compute through these singular events. As a consequence, the various coalescence modes that appear depending upon the relative ratio of the parent droplets can be studied. Computations of first stage pinch-off, second stage pinch-off, and complete engulfment are analyzed and compared to recent numerical studies and laboratory experiments. Specifically, we study the evolution of bridge radii and the related scaling laws, the minimum drop radii evolution from coalescence to satellite pinch-off, satellite sizes, and the upward stretching of the near cylindrical protrusion at the droplet top. Clear evidence of partial coalescence self-similarity is presented for parent droplet ratios between 1.66 and 4. This has been possible due to the fact that computational initial conditions only depend upon the mother droplet size, in contrast with laboratory experiments where the difficulty in establishing the same initial physical configuration is well known. The presence of electric forces changes the coalescence patterns, and it is possible to control the satellite droplet size by tuning the electrical field intensity. All of the numerical results are in very good agreement with recent laboratory experiments for water droplet coalescence.