Minecraft – Is it possible to use Minecraft bukkit AND forge at the same time

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. :)

I'm not sure if you still have this issue, but bukkit has removed wildcard support on permissions.

Bukkit has an in house permissions engine, but needs a plugin to tell it which users have which permissions. The in house permissions engine does not support wildcards and it doesn't matter which permissions plugin you have installed.

Learn to use parent/child permissions and groups.

Read http://forums.bukkit.org/threads/permissions-faq.25080/ for more details, it is under the "for everyone" spoiler.