Today's yo-ji is a great one for kanji learners, because the kanji it uses cover a broad range of commonly used words. So we can get through to the yo-ji quickly, I'll refrain from writing out the readings and definitions of the words below. Rikai-chan provides the answers faster than I can, anyhow.
喜： 喜び、喜ぶ、 喜悦、 喜劇
怒： 怒る、 怒られる、 怒り、 怒らす、 怒気、 怒鳴る、 怒髪天
哀： 哀しむ*、 哀れむ、 哀れ、 哀情(not to be confused with 愛情)
楽： 楽しむ、 楽しい、 楽、 気楽、 楽ちん、 音楽、
*This is an outdated way of writing 悲しむ ( or 悲しい and other related words). Just included these readings for the sake of the yo-ji definition.
1. The full range of human emotion
2. Joy, rage, pathos, humor
Using 喜怒哀楽 is pretty easy. You can use it as a noun, with the addition of any particle you want, but the fortunate positioning of 楽 at the end makes it easy for people to use it commonly as an adjective by tacking on ～な.
In terms of applications, you're free to use it anyway you think you can make it work, but I've found two regularly occurring instances that I can share with you:
1. 喜怒哀楽 (な or の) + (time, like 「日々、」 「瞬間、」 or 「毎日」：
This usage speaks to the human condition, the emotional ups and downs of daily life.
If you've ever heard a school related speech in Japan, you know how much emphasis gets placed on the the idea that there will be/have been "tears and laughs, and hard times, and challenges, and happy memories, and fun, and upsetting moments, etc, etc." Japanese students are inundated with the idea that the entire spectrum of feeling is a valuable part of life.
Though this particular phrasing gets used by a lot of people in writing or blogging, you can also use it when you've recently had a particularly turbulent ride on the rollercoaster of life. Have you, say, won the lottery, been dumped, finally finished that novel you were working on, and found out that your favorite steampunk necklace was radioactive in the last week? それは喜怒哀楽な一週間です。
It can be used to talk about people who are over-emotional.
The construction above is only one way to do this, and I THINK (native speaker check, maybe?) that you could probably accomplish the same feat by reffering to a person as a 喜怒哀楽な人, or by merely saying 「あの人はちょっと。。。喜怒哀楽が。。。」 Gotta love the contextuality of Japanese. It's the other person's responsibility to understand what you're trying to say, even if you don't express it completely. Or at all.
My girlfriend's emotions change so fast that it's pretty trying. One second she's mad at me, the next it's back to lovey-dovey. It happens the other way around too...