The origin of the letter is a ligature for the diphthong "OE" that has become a letter in itself. In modern Danish and Norwegian, the letter is a unique vowel, and neither a diphthong, a ligature, nor a variant of the letter "O". As one Norwegian tour guide put it, "It's not an 'O' with a slash, it's an 'Ø'." In this quotation, the pronunciation of the name "Ø" is simply the sound of the letter, which is IPA [ø].
In Danish (and more conservative Norwegian), ø is even an entire word meaning island.
The symbol "Ø" is also used in mathematics to refer to the empty set, following Bourbaki. Modern typesetting software used by mathematicians typically renders it in stylised form. For example, common TeX packages offer \\emptyset and \\varnothing, which respectively appear as:
For computers, when using the ISO 8859-1 or Unicode sets, the codes for 'Ø' and 'ø' are respectively 216 and 248, or D8 and F8 in hexadecimal. On Microsoft Windows it can by typed by holding the [Alt] key while typing 0216 or 0248 on the numeric keypad. The Unicode letter name is "Latin capital/small letter O with stroke". In HTML character entity references, required in cases where the letter is not available by ordinary coding, the codes are Ø and ø.