So long as they have a high surface to volume ratio, most plant materials from your garden will work beautifully in your compost pile. However, you want to use common sense when adding larger items like sequoia branches or giant rubber tree leaves, simply shred them or chip them up into smaller pieces - the more surface area available to you resident critters, the faster the decomposition process. Plant food scraps from the kitchen, shredded cardboard boxes, and sawdust from untreated wood will all contribute nicely to your organic potpourri, just don't put too much of any one thing in - it can throw off the N-P-K balance, not to mention the pH.
Always remember to throw in a handful of good garden soil to inoculate the new pile with living organisms. A few earthworms and rollie pollies are a nice addition, too. Though it is not required, many individuals add compost starters and accelerators to help their pile along--this is fine, just avoid the synthetic additives and seek out natural and organic sources with minimal packaging.
Though all organic matter can be broken down naturally, some materials are just not suitable for the home compost pile. First and foremost, no human or pet excrement should be added to the bins. Feces can harbor harmful bacteria, and there is no guarantee that the high temperatures of your pile will successfully kill them. Second, stay away from greasy foods, dairy products, meat scraps and bones. Not only can their decomposition result in 'colorful' aromas, they can attract rodents. Unless you are a seasoned composter, it is best to avoid them completely.
Natural chemicals in citrus peels, eucalyptus leaves, and pine needles can actually slow down your compost pile, so avoid mixing them into your artistic masterpiece. Ashes from your fireplace are basically worthless - they are already broken down as much as possible (remember the heat and flames), and therefore have little potential energy to offer the micro organisms in your pile. Beside, ash can drastically alter the pH of the soil. If your prize tomatoes experienced their worst blight on record, you probably want to keep their diseased leaves and stems out of the pile, especially if the finished compost will be returning to your vegetable garden. Why propagate pestilence? Rocks, plastic and Styrofoam are not going to do much for your creation either, so keep them out. A good rule to go by is "when in doubt, keep it out."