Q:

Is the Higgs boson an elementary particle? If so, why does it decay?

The Higgs boson is an excitation of the Higgs field and is very massive and short lived. It also interacts with the Higgs field and thus is able to experience mass. 
Why does it decay if it is supposed to be an elementary particle according to the standard model?

A:

Most fundamental particles in the standard model decay: muons, tau leptons, the heavy quarks, W and Z bosons. There’s nothing problematic about that, nor about Higgs decays. 
Your question may come from a misconception about particle decay: that it’s somehow the particle ‘coming apart’ into preexisting constituents. It’s not like that. Decays are transformations into things that weren’t there before. 

A:

Another way to answer this question is that particles are not "elementary," not even in a given quantum field theory. Quantum field theories (like the Standard Model) are expressed in terms of fields, not particles. Particles are phenomena that the model predicts; some of them are stable, some are transient (they decay). The Standard Model is constructed using an elementary Higgs field, and it predicts a Higgs particle, which is unstable.
Although the language "elementary particle" is very common and probably can't be revised at this point, it might be less confusing and more accurate to talk about the elementary fields used to express a model. Even that language isn't perfect, though, because some models can be expressed in more than one way, using seemingly-unrelated sets of fields. Quantum field theory is a rich subject with many surprises!

A:

"Decay" is just the name given to an interaction where one particle goes in and two or more particles go out.
The rule in quantum mechanics is "anything not forbidden is compulsory" – that is, any process (decay or otherwise) can happen unless it violates a conservation law. As a result, most particles, whether fundamental or composite, do decay.
The exceptions are particles for which there's literally no set of outputs you can choose that doesn't violate some conservation law. For example, electrons can't decay because the decay would have to conserve electric charge – so at least one output would have to be charged – and would also have to conserve mass/energy – so the total mass of all the outputs would have to be no larger than the electron's mass – and this is impossible because there are no electrically charged particles with lower mass. So it's stable not because it's elementary but because everything that could make it unstable is forbidden.
The muon, which is also fundamental and is almost identical to the electron except for its mass, can and does decay, because the higher input mass means that you can find outputs that conserve mass while also satisfying all other constraints.
The proton, which is not a fundamental particle, can't decay because it's the lightest particle with another conserved property called baryon number.
But, again, the particles that can't decay are the exceptions. As a rule almost everything can and does, and this applies to fundamental and composite particles alike.

