This is one of the most popular podcast players for android.

★ Dedicated search engine containing 250.000+ podcasts.
★ Minimalistic, fast and efficient; .apk size less than 350kb.
★ Audio and video support.
★ Integration with Flattr.
★ No advertising.
★ Frequently updated.
★ Automatic downloads and notifications.
★ Podcast suggestions based on current subscriptions.
★ Sleeptimer.
★ Categories.
★ Swipe navigation
★ Carmode when driving.
★ No bloat.
★ Support for Basic Http Authentication
★ Utilizes the caching mechanism of the http protocol.
★ Supported mediaformats: http://bit.ly/Cpjbh
★ OPML, RSS 2.0 and ATOM support.

If you have problems or feature requests, just drop me an email and we'll figure it out.
** Right now there are a lot of unanswered emails to which I just cannot find time to answer :/

Reviews:
http://www.theverge.com/2012/4/2/2920069/app-review-podkicker-pro
http://www.coolsmartphone.com/2013/01/30/android-app-review-podkicker-pro/

A quickstart to get you going:

The Channel tab lists the podcasts you currently subscribe to.

The Episode tab lists the 200 latest episodes from all your subscriptions.

The Downloads tab lists the episodes you have downloaded to your phone.

The rest should be quite self explanatory.