In physics, we model particles and their interactions with quantum fields. Our best current description of particle physics is called the Standard Model, and one quantum field in the standard model is called the Higgs Field. There is a phenomenon called the Higgs mechanism which means that certain particles gain mass as a result of the Higgs field. So far I have spoken in terms of the Higgs field. Particles are "oscillations" or "excitations" in a quantum field, and the Higgs boson is an oscillation in the Higgs field. It is an elementary particle (it is not made up of smaller particles, compared to say a proton which can be broken down into quarks) and it is what we call a scalar boson - a particle with 0 spin.
In the early 1960s several scientists (including Peter Higgs from whom the particle derives its name) independently arrived at a theoretical model that could explain why particles in nature have mass. To do this they realised that a new field, now known as the Higgs field, could cause particles to have mass via an effect called spontaneous symmetry breaking. The scientific community subsequently expected that the Higgs field (and hence Higgs boson) should exist. At this stage however the Higgs boson was just a theoretical conjecture - a nice mathematical trick which explained why particles have mass. To find out whether or not the Higgs actually existed, we needed to do experiments.
Creating a Higgs boson requires you to collide particles at high energy in a particle collider. When such a high energy collision takes place, a Higgs boson may be produced and this may be deduced by an appropriate detector. Producing Higgs particles (and producing them in sufficiently large numbers to make detection likely) required particles to be collided with incredibly high energies and so detailed experiments were not immediately possible with 1960s/70s technology. Serious searches took place from the 90s onwards, and ultimately the Higgs boson would be found in 2013 by the Large Hadron Collider at CERN in Switzerland.