In the paper I recently read, 'a constant pressure boundary condition' was used to impose a specific condition. The governing equations are the set of Euler equations. Does anyone hear this kind of B.C or use this? I have not heard such a B.C in compressible flow simulations. Please advice me to find out what it is.

The constant pressure boundary condition has been used when you apply the characteristic based non reflecting boundary conditions for compressible flow problems. You can find the details of the boundary condition and implementation in the paper by Poinsot and Lele (1992), Journal of Computational Physics.(I do not have a copy to send.)

Does it mean that the constant pressure condition is another name of characteristic-based NRBC?

Constant pressure condition has been used for outflow BC while characteristic based NRBC is applied for inflow, outflow, wall etc.