Very short version, as reasonably succinct as I can make it, with a minimum of technical details.
A virus is a program. In order to work its magic, two conditions have to happen: It has to be loaded into memory, and its instructions have to be executed. Some of its instructions are to self-replicate. One of the first things it may do is modify your computer’s files in such a way that the virus is now automatically loaded when you turn your computer on.
If you were to change the language around, using terms like “cell” and “DNA”, you’d be describing an actual virus: it injects itself into programs, and it (generally) duplicates itself.
Since you as a user are unlikely to just straight-up run a virus program, it has to resort to trickery. In the olden days, the virus would have to find an executable program and inject itself into the code so that when you ran the infected program, you also ran the virus.
Later on, one of the common infection vectors was from “buffer overflow attacks”. The .jpg one was mentioned above. This is challenging to explain briefly and I may need to pass over finer details.
In essence, your computer’s memory is used for both executable code and data that the executable code may refer to. A classic case is the .jpg buffer overflow. When the computer “sees” it’s to deal with an image, it will set aside a portion of memory for the image data (the “buffer”). It does this based on some data at the beginning of the file which informs the computer about the dimensions of the image, amongst other things. The viral payload sits at the end of this data, adding extra information to the image file.
When the computer loads the image data, it fills up the buffer, as predicted. The virus, however, is written “past” the limits of the buffer, into other memory that may have been used for actual running code. This can happen only because the coders who wrote the .jpg reading code trusted the data was going to be safe and didn’t enforce memory limits.
Buffer overflow attacks aren’t limited to just images. CMC’s experience above suggests it was also a buffer overflow attack (though there’s not enough information to hand to do more than guess).
In the case of your (theoretical, I hope) infected DVD, if you have all the autorun stuff turned off (I recall holding down shift while inserting media would prevent autorun) then no code from the disc would run. So long as there’s no attack vector from the file listing itself (and that looks secure), you’d be theoretically safe. If I wanted to be sure, I’d open that disc on another, older computer that contained little of value and was isolated from the network.
This turned out longer than I’d hoped. I hope I didn’t put you to sleep.