This is based on a recipe my mum used to use. She added an apple, but I prefer it without and use cider vinegar instead.
I love this with a roast, especially with roast beef.
Red cabbage changes colour based on the acidity content, so vinegar is essential to keep the deep purple colour.

This recipe is based on a smallish 1kg / 2 lb cabbage, adjust ingredients if using a larger cabbage.

Ingredients
1 red cabbage, (also called purple cabbage)
1 small onion, finely chopped
2-3 floz cider vinegar, wine vinegar can be used if adding apple
1 tblsp balsamic vinegar
2 oz brown sugar
3-4 cloves
1 small cooking apple (optional), finely chopped or shredded
1 tblsp virgin olive oil
salt & pepper to taste
dash of Worcester sauce
a little water

Method
Remove the outer leaves from the cabbage and cut into quarters. remove and discard the inner core stalk (I used to call these pine trees as a kid).
sliced and chop the cabbage into roughly 1/4" pieces.

Heat the olive oil in a large pan and add the onion and Worcester sauce. Saute gently until the onion softens.
Add the sugar, vinegars, seasoning, cloves and about 2 floz of water, stirring until the sugar dissolves.
Add the cabbage (and apple is using) and mix well.
Reduce the heat, cover pan and simmer gently for about an hour until cabbage is tender. Keep an eye on the pan and add more water if need to ensure the cabbage doesn't burn.
You can adjust the amount of sugar to your personal taste.

Can be kept in the fridge for 2-3 days and also freezes well.