Assume a system consist of a collection of particles, for example the atoms that make up a solid object.  Consider a special point, called the center of mass (CM) of the system, whose position coordinates are given below.

x_CM = Σm_ix_i/M,  y_CM = Σm_iy_i/M,  z_CM = Σm_iz_i/M.

M is the total mass of the system.

The sum is over all the particles that make up the system.
r_CM is the position vector of the center of mass.

r_CM = x_CM i + y_CM j + z_CM k = (Σm_ix_i i + Σm_iy_i j + Σm_iz_i k)/M
r_CM = (Σm_ir_i)/M.

Why do we care about this special point called the center of mass?

Let us find its velocity and its acceleration.

v_CM = dr_CM/dt =  (Σm_i dr_i/dt)/M = (Σm_iv_i)/M.
Mv_CM = Σm_iv_i = Σp_i = p_tot.

The velocity of the center of mass multiplied by the total mass of the system is equal to the total momentum of the system.

a_CM = dv_CM/dt =  (Σm_i dv_i/dt)/M = (Σm_iv_i)/M.
Ma_CM = Σm_ia_i = ΣF_i = F_tot.

The acceleration of the center of mass multiplied by the total mass of the system is equal to the total force acting on the system.

Newton's 2nd law, F = ma, when applied to an extended object, predicts the motion of a particular reference point for this object.  This reference point is called the center of mass.  The center of mass of a system moves as if the total mass of the system were concentrated at this special point.  It responds to external forces as if the total mass of the system were concentrated at this point.

The total force on a particle, F_i, is the vector sum of all internal and external forces acting on the particle.  If we sum the forces acting on all the particles of the system, then in this vector sum every internal force that particle 1 exerts on particle 2 is cancelled by the internal force that particle 2 exerts on particle 1.  This is a consequence of Newton's third law.
We therefore have  F_tot = ∑F_i =  ∑F_ext.

The total force on the system is the vector sum of all the external forces.

But   F_tot = Ma_CM = dp_CM/dt.

The total momentum of the system only changes, if external forces are acting on the system.  The center of mass of the system only accelerates, if external forces are acting on the system.
If only internal forces are acting on the particles that make up the system, then the center of mass does not accelerate, its velocity remains constant.  The total linear momentum of the system is conserved.

For the total linear momentum of a system to be constant, the particles making up the system can move with respect to each other, and can accelerate with respect to each other, but they must move and accelerate in such a way that v_CM = constant and therefore a_CM = 0.