Anything that was once living will compost, but some items are best avoided. Meat, dairy and cooked food can attract vermin and should not be home-composted.
For best results, use a mixture of types of ingredient. The right balance is something learnt by experience, but as a rough guide, use equal amounts by volume of greens and browns (see below).

Some things, like grass mowings and soft young weeds, rot quickly. They work as 'activators', getting the composting started, but on their own will decay to a smelly mess.
Older and tougher plant material is slower to rot but gives body to the finished compost - and usually makes up the bulk of a compost heap. Woody items decay very slowly; they are best chopped or shredded first, where appropriate.

compost heap

Compost ingredients
'Greens' or nitrogen rich ingredients

Urine (diluted with water 20:1)
Comfrey leaves
Grass cuttings
Other green materials
Raw vegetable peelings from your kitchen
Tea bags and leaves, coffee grounds
Young green weed growth - (avoid weeds with seeds)
Soft green prunings
Animal manure from herbivores e.g. cows and horses
Poultry manure and bedding

Browns' or carbon rich ingredients - slow to rot

Cardboard eg. cereal packets and egg boxes
Waste paper and junk mail, including shredded confidential waste
Cardboard tubes
Glossy magazines - (although it is better for the environment to pass them on to your local doctor's or dentist's surgery or send them for recycling)
Newspaper - (although it is better for the environment to send your newspapers for recycling)
Bedding from vegetarian pets e.g. rabbits, guinea pigs - hay, straw, shredded paper, wood shavings
Tough hedge clippings
Woody prunings
Old bedding plants
Wood shavings
Fallen leaves can be composted, but the best use of them is to make leafmould

Other compostable items
Wood ash, in moderation
Hair, nail clippings
Egg shells (crushed)
Natural fibres eg. 100% wool or cotton

Do NOT compost
Cooked food
Coal & coke ash
Cat litter
Dog faeces
Disposable nappies