Cosmic rays are ultra-relativistic particles traveling near the speed of light permeating the galaxy. Collisionless shock waves with their ubiquity throughout the universe and excellent capability of accelerating charged particles offer an explanation to the origin of cosmic rays. It is well established that the particles are predominately accelerated at young supernova remnant shocks through a mechanism called Diffusive Shock Acceleration (DSA). However, this theory only applies if the particles already have a relativistic starting energy. Therefore, the charged particles must be pre-accelerated up to relativistic energies by some unknown mechanism(s) before being injected into the cosmic ray acceleration process. This is known as the injection problem and a lot of effort has been put into resolving it over the past decades. This thesis will use spacecraft data from NASA's Magnetospheric Multiscale (MMS) mission to study electron acceleration at Earth's collisionless bow shock. In particular, we will study what mechanisms are able to accelerate electrons from solar wind thermal energies (~20 eV) up to mildly relativistic energies 10-100 keV. Paper III and Paper IV set out to study energetic electron events observed at Earth's bow shock by MMS. In Paper III, we investigate the most promising candidate for a solution to the long-standing electron injection problem, the Stochastic Shock Drift Acceleration (SSDA) mechanism. SSDA successfully describes a mechanism for electrons to be accelerated up to mildly relativistic energies. However, only one previous observation of the theory exists. Building on that study, we provide further evidence in favor of the theory by showing good agreement between predictions and observations. Observational evidence of an alternative electron acceleration mechanism is presented in Paper IV. The observation displays an increase in electron flux up to ~60 keV, and inconsistent features with the SSDA mechanism. The event exhibits bi-directional electron pitch angle distributions which are generally associated with magnetic bottles and are rarely observed around Earth's bow shock. The evidence led us to propose a two-step acceleration process where field-aligned electron beams are injected into a shrinking magnetic bottle configuration caused by either a shock surface deformation or a bent upstream magnetic field line intersecting the shock surface at two different locations. Papers I and II are directed more toward the heating of electrons at collisionless shocks. The studies investigate electron entropy generation at collisionless shocks and its dependence on shock parameters. Paper I states and deals mostly with the (instrumental) challenges of calculating entropy using the MMS spacecraft data. The close relation between entropy and irreversible heating is then discussed and used to classify different heating mechanisms at the shock. We show that the electron entropy generation at Earth's bow shock depends strongly on the upstream electron plasma beta and Alfvén Mach number. In the absence of collisions, the exact generation of entropy across collisionless shocks is an open question. Early theoretical studies suggest that particle-particle collisions are replaced by plasma wave-particle interaction. In Paper II, we build on the result from Paper I, by performing a statistical study of electron entropy change across Earth's bow shock and try to answer what plasma wave modes are important for entropy generation.