A discrete cosine transform is a math process that can be used to make things like MP3s, and JPEGs smaller. It does this by breaking the sound or picture into different frequencies.
One way to calculate a discrete cosine transform is to use the Fourier transformation. "Discrete" means that it works on discrete-time signals (sampled data).
For sounds, frequencies are the same as simple tones. You can make any sound by playing several tones at the same time. The discrete cosine transform is a way to find out which tones to play in order to make a given sound. The only difference between the tones is their pitch.
Human ears are good at hearing low pitches, but bad at hearing high pitches. If you use the DCT to break a sound into tones, you do not need to be as careful when playing the higher tones because people cannot hear them as well. MP3 encoders (programs which make MP3s) use this fact to make the sound smaller.

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