There's two main issues is going from a CD to headphones:
* converting from the digital form that's on the CD to analog (which is what finally drives your headphone). The thing that goes from digital to analog is called a DAC. This can be a single chip on a sound card, or complete separate box.
* amplifying the signal so it can drive your headphones
Both of these are needed whether you use a computer or not. If you use a computer, digital to analog is normally done by a DAC chip on your sound card for a computer or by a DAC in your CD player. Both of these produce "line" output, which is a signal that isn't strong enough to drive headphones. Some sound cards and some CD players have headphone outputs, but they normally aren't of very good quality. So you normally need some kind of amplifier to drive the headphones.
Some people have mentioned using a separate DAC. This is because the DAC on typical sound cards and typical CD players aren't the best quality. Most sound cards and CD players have digital outputs as well as the line outputs. The digital outputs let you plug into a separate DAC, which presumably will be higher quality than the single-chip DACs on the sound card or CD player. However the good external DACs are around $1K or more. At moderate price levels you can get good enough sound quality by choosing a good sound card or a good CD player.
Thus the simplest approach for a PC is to use a high quality sound card. There are lots of threads on choosing a sound card. Lots of people seem to like products from E-MU. Similarly, if you prefer to use a CD player, you'll choose a CD player with a fairly good DAC.
Then you will want a headphone amplifier. Again, you'll find threads discussing them.
If you are using a laptop, you can't replace the sound card. That's builtin the motherboard, and laptops don't have PCI slots for sound cards. However there are external boxes that act like act like sound cards that connect to your computer using USB. As long as you have to use an external box, you may find it convenient to get one that acts both as a sound card and an amplifier. With my iBook I use a Headroom Total Bithead. It it a pretty good sound card -- not at the level of a high-end product, but good enough for a typical user. Its amplifer will drive a headphone with moderate requirements, such as my Sennheiser 580.
I probably wouldn't choose between PC and CD player based on price. A decent sound card is probably slightly less expensive than a decent CD player, but not enough to matter. The real question is how you're doing to use it. If you just want to pop CDs in the drive, a separate CD player is simpler and probably has slightly better sound quality than the least expensive sound cards (though not by much, and you can certainly get sound cards that match a CD player). What you get from using a PC or Mac is flexibility: You can upload hundreds of CD and then use an interface like iTunes to play them. I find that a lot more convenient than changing CDs in a player. Once they're online you can also move them to an iPod or other device.
If you like the flexbility of using a computer, get a good sound card and amplifier (or a USB sound card/amp combination such as the Bithead). If you don't think you're going to use that, get a CD player and amplifier.
I have no idea why anyone is suggesting the Airport Express. If you want to play music in a different room, it makes sense. If you want to attach headphones to a laptop or desktop, the AE seems an unnecessary complication.