After a shave, the skin often needs something to recover from having a sharp blade go across it several times. Aftershaves come in various forms, the two most common being balms and splashes/lotions, balms for moisturizing the skin and lotions as astringents. Which types of aftershaves you should use varies from person to person and is to a large degree dependent on skin type.
Note that there is a difference between aftershave and cologne. While aftershave might be scented (like a shaving soap or cream) its main purpose is to be good for your skin, while it might have a lingering scent this is a side-effect and not really its main purpose.
Recommendations for various skin types: