As mentioned in the book High Fidelity, there is a certain art to making a great mix tape or CD. What songs you choose, and how you choose them, will largely affect how much your efforts will be appreciated. A compilation of songs recorded on a tape or burned onto a CD can be a thoughtful gift for someone you appreciate.

[edit] Steps

Consider your intended audience. Is this compilation for yourself? Your friends? A significant other? Select music that will challenge the listener. There's not much point in making your grandmother a CD of the best death metal from 2001 (unless she likes it, of course), but she may enjoy listening to a compilation of rare recordings of her favorite jazz artists that she listened to when she was young.
Consider whether you want the mix to convey a certain emotion or message. Pick music that you like and appreciate. How much you enjoy making the mix will be apparent in the final package.
Don't always focus on a genre and a theme. Putting widely different tracks in a compilation can add contrast for the listener.
Be flexible. Collect a set of tracks as a rough draft for the CD with the expectation you may decide not to include some of them.
Play around with the arrangement of the tracks. Think of the mix tape as a prolonged listening experience. You don't want the listener to get bored or skip songs. The first few tracks should grab the listener and get their attention. Group slower or softer songs together and then gradually build up momentum to more upbeat songs. A fast/heavy song might not go well after a soft, acoustic one.
Finalize your track arrangement and listen to the version a few times, making necessary adjustments. Feel free to remove some tracks and add others. It's possible that you may realize new tracks you'd like to add late in the process.
Burn the CD or make your master tape that you will make copies from, now that you're happy with the compilation. Try making some cover art or liner notes intended for your audience. If you've been using MP3s, keep a specific folder for your mix and name it appropriately.

[edit] Tips

With advanced CD burning software, it's possible to merge tracks, making it easy to insert sound clips (such as quotes from movies) between tracks. Merge the sound clip to the start of the track to make the CD more interesting.
Making cover art or creative liner notes can make the compilation more personal.
Picking a set of songs that really define a specific (possibly current) time in your life can be appreciated later when you put on the album and are reminded of days gone by.
It's possible to gradually build a mix CD. While listening to MP3s, if you come across a song that would be a good fit for a compilation, copy it over to a folder reserved just for your ongoing compilation.
Come up with a clever or thoughtful title for your compilation to make it more memorable.
The last song you put on the compilation is always important. What message do you want to leave with the listener? Do you want to end it with a bang? Or would you like to end the compilation slowly and softly? Tying in the last song to the theme of the compilation can make it much more effective.
Consider using software such as Ableton. It's NOT cheating and will allow you to make your mix far more interesting by adding effects and looping sections as you see fit. It's also a useful tool to get your head around should you decide to mix on the fly.

[edit] Warnings

Watch your attitude! Be certain you are in the mood you wish the CD to convey when you make the mix CD -- otherwise, other feelings are likely to seep into the songs you choose.
Avoid picking several songs from one artist. Rather, focus on a wide range of artists. Especially try to avoid including two songs by the same artist back-to-back. Of course, there are always exceptions, such as songs that are made to be played together (such as "The Hellion" and "Electric Eye" by Judas Priest, "We Will Rock You" and "We are the Champions" by Queen, or "The Load-Out" and "Stay" by Jackson Browne) and two songs that have special meaning to your intended audience when played together.
Make the compilation running time no longer than necessary -- if possible, keep it under an hour. Weed out any songs that don't fit. If the CD is too long, it's possible some great tracks near the end will never be heard.
There's really no such thing as the definitive, be-all, end-all, perfect mix CD. The guidelines established here are things to keep in mind to help you mold your CD into being, not a list of rules that are meant to be strictly adhered to. Play around, try new things, be creative, but always take your audience into consideration or all of your hard work will be for nothing!
You are using some one else's words and feelings to express your own, and you must be very careful with this. Pay attention to the lyrics of the song so they go along EXACTLY with the message you are trying to convey.