There are a lot of language learning podcasts out there and I'm sure you can find at least a couple of good ones about your target language, but did you know that's just the tip of the iceberg? There are other ways to learn and practice your target language using podcasts, funnier and even more effective ways. We will get to them in a second, but first let's just look at what kind of podcasts you should be looking for.
Topics that you are passionate about
Would you ever listen to something in your native language that you have no interest for? I'm guessing the answer is no. Then there's no use in doing that with your second language either! Pick a topic that you like and search for podcasts about it in your target language. For example, if you want to learn French and you love cooking, just type "cuisine" or "recettes" in the iTunes search tool. This way not only will you listen to something you're passionate about (and maybe even learn something new), but you will also learn terms specific for that particular topic, which is awesome because then you will be able to have a conversation about that subject in your target language!

Kids stories
They will generally contain easier terms and expressions. Sometimes you can even find free audiobooks around the internet.

This kind of audio can be very difficult to understand, but I included it because I think it's quite important for intermediate and advanced learners to become familiar with the way people talk in the news. The good thing about it is that the speaker will talk very clearly and without slang or accents so even beginners can take advantage from this.

Comedy shows
Laughing is good for your heart and it also prevents boredom! What's better than learning new words while having fun?

Anything where people talk slowly
Audio alone is generally more difficult to understand than video; that's why it's nice to have someone speaking slowly and clearly. I've noticed that people in podcasts about fitness, health, and especially breathing exercises tend to speak more slowly than usual, so you might want to give those topics a try.

Now that you have a general idea of what to look for, let's look at 4 ways you can use all this material for language learning purposes:
1. Background listening
Every time I am not doing something that requires me to think too hard or interact with others, I listen to a random podcast; 70% of the time I am not paying full attention to what I hear, but I am letting my brain get used to the sounds, inflections, rhythm and pauses of the language. Every once in a while I listen more carefully, but eventually drop back into this "unconscious listening" state. Getting used to the sounds of a language is really important, it will help you sound more natural and fluent when you're speaking. Some people even listen to podcasts while they're asleep!

2. Shadowing
For those who have never heard of it, shadowing is the technique of repeating words that you hear almost simultaneously, mimicking every sound as well as you can. Shadowing can give a boost to your pronunciation as well as your fluency. When you're listening to a podcast, try some shadowing and see how close you can get to sounding like a native speaker. Some podcasts also have pdf transcripts which are very helpful because you can use them to follow along with more ease.
3. Learn new terms
Sometimes I play a little game that I like to call "podcast apocalypse": I listen to a podcast (usually no more than 4-5 minutes long) and try to write all the words that I don't understand on a piece of paper. Then I look them up in a dictionary (if you're intermediate/advanced learners I recommend a monolingual dictionary) and save them in an SRS like Anki for future review. Then I listen to the podcast again and if I did a good job, the second time I will magically understand most of it! It's a fun and gratifying way of learning new terms.
4. Create your own language learning podcast
You've been listening to podcasts about -and in- your target language for so long that you might as well start your own! Why don't you let the world know that you're studying a language and how much effort you're putting in it? Just start a blog about learning that language and then record video or audio podcasts where you explain every new thing you've learned, try to say something in that language and share the material you are using for your studies. You may get to know other people who are learning the same language and share tips and tricks with them. 
The best thing about this is that making your progress public for virtually the whole world to see will really boost your motivation.
This is maybe the most "extreme" way of using a podcast for learning a language and the one that would require the most time and commitment, but it can also be highly rewarding and if you like blogging and have some time to dedicate to it, definitely give it a try!

I hope this list was useful and inspiring for all you language learners out there. Now head over to iTunes and start looking for some cool podcasts to listen to!