Technically, still hasn't ended. In parts of the world without reliable roads/repair facilities/fuel, horses/other riding animals are still used. As far as Western and European armies are concerned, there were horse cavalry units deployed through the end of WWII. The effect of the cessation would be that the requirement for veterinarians in the military has dropped drastically, use, purchasing and storage of feed has stopped, etc. etc.

The battle of Beersheba in WW1 was the last successful cavalry battle. Later in WW2, the polish deployed lancers against panzer regiments and were obviously flattened. My guess would be that as projectile weapons became more effective, mounted troops became less so. During the Hundred Years' War, Longbowmen defeated mounted Knights. Longbows had pretty good range, knockdown or penetration power and a good rate of fire. You can only armour a horse so much, before it loses mobility and practical use. Horses are also a larger target than a person and harder to conceal or cover. Flintlock weapons had a decent range (rifles and artillery) and good knockdown power, however their rate of fire wasn't sufficient to render cavalry obsolete. It wasn't until more automatic guns matured that cavalry became completely obsolete.

Edit: I am primarily talking about cavalry where fighting occurs on horseback. Horses are still useful as transport where terrain or stealth needs prevent normal land vehicles from operating. Also Beersheba was won by the Australian Light Horsemen, who were created primarily as dragoons. However like dragoons they often fought on horseback where circumstances required, such as the charge on Beersheba.