It really depends on how vermin proof your composter is and whether you have extra worms and stuff like that. For an ordinary composter, stuff like leftover steamed or boiled vegetables are no problem at all; starchy stuff like potatoes or yams you'll get away with in small amounts occasionally (but feed the birds with your stale bread, don't stick it in the composter). Unless you really really know what you're doing and have suitable hardware (I don't!!) trying to compost fried foods / dairy / meats, fish, eggs is very high risk for bad odours and vermin problems - not to mention unusable slime at the end of the process.
I think there was a sort of super composter called "Big Bertha", if memory serves, which was supposed to be able to deal with virtually anything.