If you want to understand the mechanics of the probing system, you need to read Hoshi's guide on probing which can be found here. For the purposes of this guide I'm just going to cover the practical aspects as they relate to exploration, and skip as much of the theory as possible.
Introduction
The Revelations expansion (codename Kali) introduces proper exploration to Eve. Over 250 individual exploration sites have been created and added to the game, offering a wide variety of content. Each site is unique, but there are four broad categories they fall into - combat sites, profession sites, mining sites and gas cloud sites. 

Sites are distributed randomly all over the Eve Isk cluster, in all security statuses and regions. They appear in planetary gravity wells, stick around for a while and then disappear again. As such sites are not permanent enough to be documented, you have to track them down yourself using the exploration tools if you want to make use of them, and it's these tools that will be explained in this guide, along with some notes on how to deal with the sites themselves.

Cliff Notes mechanics
Exploration is done using scan probes. Changes to probes in Revelations make them all chance-based, and that applies to exploration too. You'll need to get your equipment and skills together, and load up with the right probes. You use a Multispectral Probe to check if there's anything in the system worth looking for and, if there is, launch a Quest probe around each planet. You then analyze them all together, over and over again, until the random number generator spits out the right number and you get a result. Once you get your Eve Isk first result, you can use more accurate probes to zero in on the site, eventually dropping you right on top of it. Then you just pew-pew the bad guys and apply whatever tools are necessary to plunder your site.

Some things to note:
1) Exploration is chance-based. You will often need to run a lot of analysis cycles before you find anything
2) Exploration uses a fair number of new skills. It's worth training up and getting prepped first
3) Exploration rewards people who know what they're doing. So read the guide careful!

The Walkthrough