Elastic means kinetic energy is conserved.
Inelastic means it is not, some energy is lost in the noise, crunch, etc.
Totally inelastic is a special case of inelastic, where you have the extra information that the two objects stick together after the collision.
Totally inelastic is the easiest to solve, because there is only one v after the collision.
Elastic is harder, because you have two equations for two velocities: conservation of momentum, and conservation of energy. In fact, a lot of people find the quadratic equation that results from substitution to be a pain, so there is actually a shortcut that works ONLY for elastic, head-on (i.e. one dimensional) collisions:
v1 + v1' = v2 + v2'
So if you combine that with
m1v1 +m2v2 = m1v1' +m2v2'
You get two easier equations in two unknowns.
I tutor physics.