Building your website on a shared server means that your pages may be affected by a neighboring site that devours too many server resources. For example, if that site receives a huge spike in traffic, your pages might load slowly—or not at all. Investing in a dedicated server greatly reduces this potential problem, plus it gives system administrators greater control over the apps and scripts that they can install on the server, too. Shared hosting is far more limited when it comes to what you're allowed to do, because everything you do could potentially affect the other sites with which you share the server. When you've got the server all to yourself, your scripts and apps won't impinge on anyone else's bandwidth or RAM.
A gaming server is a dedicated server used for online games such as World of Warcraft or Minecraft. The benefit of a dedicated server for gaming is that your server is often more stable than the game client itself, which is impacted by multiple players and resources. For a smooth gaming experience that you control, a dedicated server is the way to go.