For an incompressible flow:

\nu d^2 u_i / dx_j dx_j - Viscosity

-1/\rho dp/dx_i - Pressure gradient

u_j du_i / dx_j - Advection. Kinda like momentum transfer from the motion of the fluid itself. Nonlinear, which makes the N-S equations hard to solve

du_i/dt - Rate of change of velocity. Note that this is in an Eulerian framework so it's not the acceleration of a packet of fluid, rather it's just the change in velocity at a particular location in space

Euler is when you omit some terms. Forcing is when you add some other terms to account for phenomena external to the fluid like gravity or flow through a porous medium like in the article.