There are around 400 billion star systems in Elite Dangerous and currently less than 0.01% has actually been explored.  That's including all players from PC and consoles.

Exploration gives players the opportunity to earn credits by using a Discovery Scanner to scan star systems and using a Planetary Surface Scanner to scan individual stars and planets.  Players can then sell the data collected at ports for credits.  If you manage to scan any stars or planets that haven't previously been scanned by anyone else you not only will get a bigger payout but also have your name (gamertag, PSN, etc) displayed on the planetary body.

I'm not going to tell you all the ins and outs of exploring.  If you'd like to know more about exploring in Elite Dangerous, there is an excellent wiki page with all sorts of information about exploring including tips for building exploration ships, guides for how to set up the star filter in the galaxy map, how many credits you'll earn for scanning each type of planetary body, etc.

Technically there's no such thing as an exploration ship, all ships in Elite Dangerous can be fitted out for exploration.  The only thing is; some ships are better suited than others.
Below is a table of all ships currently in Elite Dangerous showing both unmodded and modded jump range and price.  As you can see, if you're strapped for cash, you can still have a reasonable exploration build for a Hauler, but, if you can afford it I recommend the Diamondback Explorer because it has the largest unmodded jump range, it's relatively cheap to buy and build, and because of its small size you can land at any port.

Note:
The prices in the table below were taken from the Coriolis ship builder website and will vary at different ports across the galaxy, e.g. Li Yong-Rui controlled systems offer a 15% discount and in Jameson Memorial it's 10%.
Also, the builds are based on pre-update 2.4 modules so do not include repair limpets.  I am waiting for an update to the production Coriolis website and the Beyond 3.0 before I redo all the ship builds.

Note:  The engineer changes in the Beyond 3.0 update will change some of these figures but you'll still be able to use it as a guide

Note:  The ship builds in this list are subject to change