1. Regular old soap works by causing bacteria on your hands to loosen their grip and/or removing whatever the bacteria are clinging to.  In other words, regular soap doesn’t actually kill bacteria, it merely removes it in conjunction with scrubbing.
2. SCRUB SCRUB SCRUB: To have an effective hand washing you need to scrub your hands with soap and water for at least 20 seconds. The soap is not a magic elixir that you merely rinse over your hands. 
3. No soap?  Scrubbing your hands for over 20 seconds with just water is still somewhat effective at loosening bacteria and cleaning your hands.
4. Antibacterial soap, unlike regular soap, does kill bacteria and is more effective at cleaning your hands than non-antibacterial soap.
5. However, antibacterial soap likely damages your microbiome (the trillions of bacteria living mostly in your gut but also on your skin and elsewhere), can lead to antibiotic-resistant bacteria and is thought to cause more harm than good.
6. The FDA recently banned antibacterial soap.  In doing so it stated that regular soap and vigorous hand washing works very well and that there is no evidence that antibacterial soap reduces disease or infection.

Sources: https://www.theatlantic.com/health/archive/2012/06/soap-how-much-cleaner-does-it-actually-make-your-hands/258839/  and http://www.npr.org/sections/health-shots/2016/09/02/492394717/fda-bans-19-chemicals-used-in-antibacterial-soaps