The easiest way to describe it as that all network traffic is treated the same by internet providers.
When you access
www.sherdog.com, based on your ISP, the Sherdog servers and connection, it will load without restriction. At the same time, a major site like google.com or even a some guy's amateur porn site load and operate the same way.
Without net neutrality, ISPs would have the ability to speed up or slow down certain websites or block certain websites or content. It would open up a situation where Netflix could pay millions of dollars to Comcast in order to have faster service for their website. It could also allow Comcast to slow down certain undesirable traffic such as Bit Torrent.
Basically Net Neutrality keeps the internet the way it is now and protects it with law. The cause of concern from the Right and Conservatives is that they don't want the government/FCC regulating the internet and they feel that it's unnecessary and prevents growth.
The truth is that unfortunately we DO need regulation because due to no laws being in place, ISPs and corporations have tried to sue the FCC in order to allow them to slow up and speed traffic as they see fit and allow for companies to pay them to do so.
Net neutrality is a good thing and thankfully it was passed today. There's a bunch of fear mongering out there from a lot of people that do not understand the internet or technology. All you have to do is look at the companies and people that support Net Neutrality vs. the ones that don't and it's clear on who you should trust.