A Private Server is operated by private parties or volunteers. For some games, people make private servers so that they and others can play for free but that's not the case with Maplestory because it's already free to play. Still, MapleStory private servers exist for several reasons.
A private server is illegal and playing on them is usually frowned upon. Most private servers are really unstable and full of bugs so the game experience is not as high in quality as the official servers. Of course, some people still like playing on them because they can use bots, macros, hacks and cheats without worrying about getting banned.
If you are looking for a private server to play on or want to find out how to start your own server, then check out MmorpGuides.com. They have a forum with a lot of friendly and knowledgeable members who help each other out. They'll know if there are any really good Maple Story Private Servers out there.