Many folks across the United States are covered in snow right now - even folks who are not used to getting a lot of snow.
Welcome to Nate 90!
The word "podcast" is like the snow across our country right now - it's everywhere and many folks don't know what to do with it.
Over that last year, I have been ramping up this podcast as well as my personal blog. In that process, I have been sharing episodes of this podcast on my blog, and some folks in my life who haven't ever thought about podcasts, are now hearing about them from me.
So, I thought I would answer the question:
What is a podcast?
The simple answer is this:
A podcast is primarily an audio only, radio-like show with semi-regular episodes that is primarily listened to through a smartphone. 
Listening to a podcast is like listening to some sort of non-music radio show, including all kinds of topics. Any sort of radio show you can think of likely has a podcast which covers the same topic.
The most significant difference that I can think of between a radio show and a podcast is that of accessibility. In order to listen to a radio show, you have to be within range of the radio station (unless you are listening to satellite radio) and tune in at a specific time. A podcast is available on demand and is available anywhere an internet connection is available.
I did a little bit of research on the history of podcasts, and I was surprised at all the details available.  I had assumed that podcasting was mostly an Apple iTunes thing because of their one-time popular iPod's and the iTunes store. I remember in 2008-2009 starting to listen to podcasts in the iTunes program.
As with many modern technological tools, the roots of podcasting find their way back to the 1970s and 80s, slowly gained momentum, and then, finally took off in the mid-2000s.
When I was a child in the 1980s, you were limited to how many radio stations were available, and some of us to how many TV stations were available too (we never had cable TV). Even then, those who had cable TV, were limited to a certain number of channels. I remember going over to friends houses and watching TV. If you watched enough of it, you began to see repeats.
With on demand podcasts and on demand TV, there are an overwhelming about listening and watching possibilities available anytime of day. Because of this, some people may wonder, why bother? It's information overload! And, that is understandable.
Here are some possible benefits of on demand podcasts:
And last of all, I don't want to neglect the most practical part of a podcast - podcast players. Smartphones have apps that are used to subscribe and listen to the podcasts. Usually, when a new episode from one of your podcast subscriptions comes out, your podcast app will send you a notification. Also, many of these apps allow you to create a listening list along with many other common features.
A couple popular podcast apps for the iPhone are the Apple Podcasts App and the Overcast App. For Android, Stitcher, Google Podcasts, and Spotify.
There are several podcast directories out there, and these Apps will usually search them (I think the Apple Podcasts App only searches the iTunes Directory, but you do have the option of manually entering the address for a podcast not listed in iTunes).
 This is my definition, and I am not aware that I am quoting someone else's.
 See the Wikipedia article: https://en.wikipedia.org/wiki/History_of_podcasting