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. Aftershaves often contain an antiseptic agent such as denatured alcohol or stearate citrate to prevent infections from cuts, nicks, and weepers. Some aftershaves also have menthol as an ingredient to help cool and numb damaged skin.
Recommendations for various skin types: