The ability to measure and manipulate complex particles in the solid state is a cornerstone of modern condensed-matter physics. Typical excitations of a sea of electrons, called quasiparticles, have properties similar to those of free electrons. However, in recent years exotic excitations with very different properties have been created in designer quantum materials, including Dirac fermions in graphene [1] and fractionally charged quasiparticles in fractional quantum Hall systems [2]. Here we report signatures of a new quasiparticle -- the Majorana fermion -- in Josephson junctions consisting of two superconducting leads coupled through a three-dimensional topological insulator [3]. We observe two striking departures from the common transport properties of Josephson junctions: a characteristic energy that scales inversely with the width of the junction, and a low characteristic magnetic field for suppressing supercurrent. To explain these effects, we propose a phenomenological model in which a one-dimensional wire of Majorana fermions is present along the width of the junction, similar to a theoretical prediction by Fu and Kane [4]. These results present an opening into the investigation of Majorana fermions in the solid state and their exotic properties, including non-Abelian statistics [5], a suggested basis for fault-tolerant quantum computation [6]. Show Partial Abstract