Here's the ones I use, from the most used, to the least:

    - EDICT, for quick lookups, checking up readings;
    - 研究社 新和英辞典, for word usages, sample sentences;
    - 英辞郎, for slang, idioms, tricky sentences;
    - 大辞林, for pitch accent, mostly;
    - 広辞苑, for more in depth explanations.

EDICT is available at http://jisho.org, 英辞郎 is available at http://www.alc.co.jp/, and 大辞林 is available at http://dictionary.goo.ne.jp/. The other two are paid dictionaries, but I think they're well worth it.

大辞林 is super pimp. It is my main, and most loved, dictionary. I suggest everybody use it.

I recommend 研究社's 新和英大辞典 if you want a J-E dictionary. I posted the reasons why I like it in another thread.

I've never seen a good E-J dictionary for Japanese learners. The only E-J dictionary I think might be helpful is 大修館's ジーニアス英和大辞典. It is designed for English learners and explains in great detail word usage and subtle nuances the Japanese would often have trouble understanding, so you might be able to learn differences between English words and their equivalents in Japanese. For example, the entry for "borrow" explains the difference between "borrow," "use," "hire," and "rent" because English learners often mix them up, so you can learn 借りる covers all the meanings. I never ever recommend it for sentence mining, though.

As for J-J dictionaries, I think 明鏡国語辞典 published by 大修館 is the best. It's a small dictionary for native speakers so obscure words are not included, but it focuses on contemporary word usage and differences between synonyms, i.e., how to use basic words. Learners often ask questions like "What's the difference between XXX and YYY?" and "When do I kanjify the word XXX?" This dictionary often gives answers to such questions. Actually some of my posts about Japanese word usage are simply translations of entries in this dictionary. If you want to learn the difference between 会う, 逢う, 遭う, and 遇う, all of which are あう and mean "meet" or "see" in English, then this is the right dictionary. Want to know when 辺り is written in kana? Look in this dictionary.

As some posters already said, 大辞泉 and 大辞林 are really good J-J dictionaries, too. But I think they're assuming the reader has a good command of Japanese; you should already know 暗くなる前に and 暗くならない前に both mean "before it gets dark." If you thought they meant the opposite because the latter is grammatically negating the former, go get 明鏡国語辞典 and look up 前. Neither 大辞泉 nor 大辞林 explains this because it's obvious to native speakers, though English speakers may wonder why ない doesn't negate the sentence.

That said, the dictionaries I mentioned are all for native speakers, so don't expect too much. I think the lack of decent J-J dictionaries designed for learners is the biggest obstacle. Every famous dictionary publishers in the UK and the US publishes excellent E-E dictionaries for learners, but Japanese publishers don't give a shit about foreigners...