You'll need an RCA-to-minijack plug if you want to use speakers. These things are cheap at RadioShack or other similar electronics shops. If you don't have one you can buy online too.
You'll probably want to use some sort of amplifier if you're using a headset. If your monitor has built in speakers (and a built in headphone jack), you can connect your 360 to the monitor speakers and then plug the headphones into the monitor. If your monitor doesn't have speakers/doesn't have a headphone jack, then you probably already have some PC speakers that do have a headphone jack, in which case you want to connect the speakers to the 360 (and plug the headphones into the speakers).
Here's a simple diagram of some options:
All those parts are available at RadioShack. If you can't find a coupler, you can use a stereo headphone splitter (the thing that lets you connect two headphones to one device) and just use both female ends (put tape around the male end to avoid static causing noise if you must).
If you're using the speakers (method 2 in diagram), check to see if your speakers can accept an RCA input. Some speakers can use the red/white plugs, others only use minijack (headphones plug connector). Also, some speakers have a female connector for the input while others have a cord with a male connector (usually the green plug as shown in the diagram). If you have a speaker set with a female input, you don't need a coupler or splitter.
If you don't have speakers OR a monitor with built in speakers, you can try method 3 but I doubt it'll sound very good if it even works. For this you should either pick up a cheap set of speakers (they'll have a headphone amp built in usually) or build one of the numerous headphone amplifiers that you can find online.
