THE STANDARD MODEL (SM) of Particle Physics is the dynamical theory of strong and electroweak interactions. It is the distillation of all that is known about the interactions of fundamental particles and has proven to be stupendously successful by surviving every experimental test performed to date, apart from some important aspects of Neutrino Physics (see Neutrino Physics article). Yet one day it will surely fail.
Why should we expect this? There are two principal reasons. Firstly, a major component of the SM called the "Higgs boson sector" has yet to receive direct experimental support. If future experiments should disconfirm the Higgs boson hypothesis, then the SM will require radical reconstruction. Secondly, the SM is not able to explain many fundamental features of the elementary particle spectrum. For instance, all types of quarks and leptons are observed to be replicated three times (for example, the electron is joined by the muon and the tauon). As far as the SM is concerned, this number could have been any positive integer, including zero! What is special about the number three? Another deep mystery is the origin of quark and lepton masses. In the SM, all of these masses are arbitrary parameters whose precise values cannot be explained but must be measured. In order to understand the universe properly, the problem of the origin of mass must be solved. Think, for instance, of how different chemistry would be, and hence conditions in the whole universe, if the electron had a mass comparable to the proton. On these and other issues, the SM is silent and therefore presumably incomplete.
The study of Physics Beyond the Standard Model is in large measure an exploration of symmetry. The SM is founded upon certain mathematical transformations, which leave the fundamental equations of motion of the theory invariant. Such transformations are called "symmetries." One hopes that by enlarging the group of symmetry transformations, the equations of motion of Nature become more constrained, and so many apparently arbitrary features of the SM get interconnected. Ultimately, a theory might be so constraining as to not admit a mass for the electron that is different from the one observed experimentally!
Speculations about Physics Beyond the Standard Model also bear on Cosmology (see The Early Universe article): The origin of the matter-antimatter asymmetry of the universe is almost certainly tied to new physics (see B Physics and CP Violation article). The production of exotic objects such as Cosmic Strings and Magnetic Monopoles is associated with symmetry breaking. The physics of neutrinos, for which there is already strong evidence of non-standard behaviour, affects some light element syntheses during an epoch soon after the Big Bang. The dark matter of the universe could well exist in the form exotic particles.
We hope to understand so much more, once we know how to go... Beyond the Standard Model!