The MOHO is the name given to the boundary between the crust and the mantle. It's existance was initially detected through a discontinuity in the motion of seismic waves from one point to another through the Earth. Motion of the waves along the MOHO is faster than in the crust itself. So beyond a certain seperation it is quicker for the waves to descend to the level of the MOHO, travel along the MOHO and then return to the surface than it is for them to move along in the crust.
The boundary is really sharp. Evidence for this is seen in the seismic data itself as well as inspection of parts of the MOHO that were deposited in Oman during obduction of an oceanic plate.
The difference in speed for the waves represents a difference in refractive index and this is obviously associated with a difference in compositon. Whilst the crust is made up mainly of basalt and granite the mantle is a peridotite based rock, featuring olivine, and some pyroxenes.
The lithosphere is not to be confused with the crust. The crust is only part of the lithosphere. The lithosphere extends to approximately 100 km in depth and it is the fracturing of the lithosphere into plates that is described by plate tectonics. The lithosphere therefore includes the MOHO - the boundary between the crust and the mantle.
The asthenosphere and mantle are not the same thing. The asthenosphere is the majority of the mantle but not the uppermost section which is part of the lithosphere. The distinction between the lithospheric mantle and the asthenosphere is not one of composition but to do with the way it is able to deform. The asthenosphere can flow by diffusion creep whereas the lithospheric mantle can only move by dislocation creep. The result is that the asthenosphere can (and does) undergo convection which keeps it's temperature more or less constant and the lithosphere has a sharp temperature gradient.