Minecraft – How to teleport to other bukkit servers without disconnecting via the pause menu and filling in the new IP

Bukkit is a developer API - it basically provides hooks and functions that developers can use to plugin to Minecraft.

Craftbukkit is the server component you download that lets these plugins work with the official Minecraft server.

Craftbukkit is not yet a package manager, rather a server wrapper - plugin installation and managed updates are planned through another tool (fill.bukkit.org) but it has not yet been released - Bukkit itself is still under heavy development.

So, to answer your three bullet points:

What Bukkit does that vanilla Minecraft Server does not

Bukkit on its own does very little – it simply exposes an API that can be used by developers to add functionality to the game.

Why people decide to use it over vanilla Minecraft Server

People generally use it because of the additional functionality that can be added through plugins: area protection, a form of economy, the ability to warp throughout the world, and making minecarts more stable. A quick browse through the Bukkit plugin releases forum will show you some more possibilities.

What plugins do/are for

I think this is answered through the other two – they add additional functionality to the game. :)

This is a glitch with Minecraft that it also spawns multiple pigmen from one pig. The easiest way is to disconnect and reconnect. They should disappear.

This has happened in my server.

And it IS a client side bug like SevenSidedDie said.