What is Disc Golf? It is the game of golf where discs replace both the club and ball of traditional golf. The object is to get your disc to the target in as few throws as possible. The discs used for the game are similar to frisbees in some sense but they are bit smaller and many have an almost sharp edge. The discs are typically classified as Drivers, Mid-ranges, and Putters though any disc can be used for any throw – you could tee off with your putter and putt with your driver if you like (I actually did just that at the Philly open in 2015). 
You putt into a metal basket with chains that catch the disc. The game is played on a course that usually features 18 holes, just like traditional golf. Some are smaller 9 holes courses. The courses range from beginner friendly pitch & putt style to pro championship caliber locations that challenge the very best disc golfers in the world. It’s free (or really cheap) to play, and the discs are inexpensive so unlike traditional golf it really is accessible to the masses. That’s it in a nutshell, but here’s some more info if you want to dive deeper:
What do I need to play? About $25 for Discs, usually you want to pick up a Mid-Range and Putter if it’s your first time out. 
After you tee off you always throw from behind where your previous shot came to rest so two discs is pretty ideal. Most starter packs have three discs; a Driver, Mid, and Putter. Anyway go to your local disc shop and buy them or if you gotta use the innernets try these places:
Keystone Disc Golf
Marshall Street
Disc Golf Center
Infinite Discs
How do you play this game again? Just like regular golf each hole has a par score – usually three throws. 
A bogey is one over par, a birdie is one under par etc. You play eighteen holes and add the total number of throws it took to complete the course, lowest score wins. The complete rules are HERE. Also, there are lots of variations of the game besides lowest score. You can do Match Play, Skins, Doubles, Wolf etc. etc.
Where and When can we play? Good news, there are as many disc golf courses in America as traditional ball golf and we’re growing fast. Also there are no tee times, you’re free to play when you want to for the most part. Check these resourses:
Disc Golf Course Review
Disc Golf Scene
Who’s in charge here? Well, most courses are maintained by local leagues in coordination with the parks and rec staff. Check out Facebook or search the web for local league info. Lots of volunteer hours go into making a disc golf course clean, playable, fun and safe. If you find yourself on a nice course some time, you should know a lot of folks worked hard to make it that way. Respect the course, respect the effort. Oh, also the competitive sport of disc golf is governed by the PDGA – you’ll figure out what it stands for. That site has the official rule book in pdf if you need it. If you decide to play tournaments you’ll want to join, there is no downside to becoming a member.
Why should I play Disc Golf? This is where I wax philosophical about how it’s healthy, it’s fun, it’s affordable, it’s good for families with kids as well as middle age guys like myself or old farts like my brother. I dunno, it is all that stuff and all of those are great reasons to play. 
For me, I love the flight of the discs. Being able to control their flight by manipulating the release and spin out of your hand is pretty magical and very addictive. Courses are located in some of the nicest parks in my area, exploring the outdoors and soaking it all in is living-right in my book. Anyway, I got hooked on the game then I got hooked on the sport. It’s been a life changer. I go to bed early, I eat healthier, I’m more social and active in my community, all because I want to be a good disc golfer. Beyond throwing a disc well, a good disc golfer is someone who represents the game in a positive way, cares for the courses and the local leagues and helps introduce people to the game and grow the sport. I’m trying to do all those things and little by little it’s happening for me. Give it a try and you might find your own reasons.