We perform a numerical-relativity simulation for the merger of binary neutron stars with 6 nuclear-theory-based equations of states (EOSs) described by piecewise polytropes. Our purpose is to explore the dependence of the dynamical behavior of the binary neutron star merger and resulting gravitational waveforms on the EOS of the supernuclear-density matter. The numerical results show that the merger process and the first outcome are classified into three types: (i) a black hole is promptly formed, (ii) a short-lived hypermassive neutron star (HMNS) is formed, (iii) a long-lived HMNS is formed. The type of the merger depends strongly on the EOS and on the total mass of the binaries. For the EOS with which the maximum mass is larger than 2M(circle dot), the lifetime of the HMNS is longer than 10 ms for a total mass m(0) = 2.7M(circle dot). A recent radio observation suggests that the maximum mass of spherical neutron stars is Mmax >= 1.97 +/- 0.04M(circle dot) in one sigma level. This fact and our results support the possible existence of a HMNS soon after the onset of the merger for a typical binary neutron star with m(0) = 2.7M(circle dot). We also show that the torus mass surrounding the remnant black hole is correlated with the type of the merger process; the torus mass could be large, >= 0.1M(circle dot), in the case that a long-lived HMNS is formed. We also show that gravitational waves carry information of the merger process, the remnant, and the torus mass surrounding a black hole.