Podcasts are my primary source of information. I still listen to radio news and use a lot of web sites but I think if I added up the time I spend on these different media I would find podcasts win easily. Apple do provide a podcast app for free with iOS but many people (including me) don't like it much and have invested a small amount (NZ$2.59) in a more capable alternative, such as Downcast.
So a short primer on podcasts. These are audio (mainly, although some also include video and still photos) programs on a huge variety of topics: news, commentary, tips, entertainment, etc. They are downloaded from the internet and are produced on a regular schedule (usually). New podcasts are listed in feeds which podcast programs can read and know when new material is available. Usually the new program is stored on the iPhone (or iPad or other device) but some programs also allow streaming. The user can then listen to the podcast when they are ready.
The list at the left shows the title screens of the first few podcasts in my list. On the right are details of the last 2 episodes of the elected podcast: The MacCast. At the top are the controls for playing and working with the individual episodes. This shows the iPad screen but I usually use the more portable iPhone to listen to podcasts while walking around and driving.
Apple really created podcasting but have neglected it a bit recently. The Apple podcast app was originally rather slow and awkward, was a bit buggy and was missing some useful features, plus had a horrible skeuomorphic design (what were they thinking?). But apps are cheap so I moved to Downcast and have never regretted it.
Downcast fully supports all the special podcast features such as embedded graphics, chapter markers, links, and text. It keeps subscriptions and episodes synced across multiple devices. It has great configurability on an individual subscription basis (for example, I have the longer podcasts, like the MacCast stop after each episode so I can choose a new topic, but shorter ones, like the Radio New Zealand news just plays all the episodes I have sequentially). Plus it has a clean, functional design.
If you are serious about podcasts or even just use them casually, and find Apple's podcast app frustrating, then Downcast is definitely worth the price of a cheap coffee!