1. Solid Rocket Boosters (use solid fuel chemical propulsion)
2. Liquid Rocket Boosters (use liquid fuel chemical propulsion)
3. Cold-Gas Rocket Boosters (use cold-gas chemical propulsion)
4. Ion Thruster (use electric propulsion )We will concentrate on the ion thruster in this article.
The Ion thruster creates thrust by accelerating ions with electricity, some of its advantages are Restartable, unlike the solid rocket boosters, once you start it you can't shout it down,
Controllable can be controlled easily, High Specific Impedance, which means but in an electrical wise that a point in an electrical circuit allows a relatively small amount of current through, per unit applied voltage at that point. High impedance circuits are low current, high voltage.
but one of its disadvantages is its complexity.
Ion engines are vastly different from chemical rocket boosters (solid, liquid) engines in that they are low thrust engines which can run for an extended period of time. the length of use of chemical engines is usually from seconds to days while the length os use of ion engines can be anywhere from days to months.
For even more specific details please visit this link:
http://www.qrg.northwestern.edu/projects/vss/docs/propulsion/2-what-are-the-types-of-rocket-propulsion.html
https://en.wikipedia.org/wiki/Ion_thruster#/media/File:Ion_Engine_Test_Firing_-_GPN-2000-000482.jpg