What's the difference between the words "plate" and "dish"?

I'm confused about the words dish and plate. Could someone please explain the difference?

A plate is a specific type of dish. A dish is generally something that food is eaten or served from. A plate tends to refer specifically to a flat dish that is suitable for holding food that does not have a high liquid content. The other common type of dish is a bowl. A bowl will be similar to a hollow sphere cut in half and is useful for serving soups and other foods with high liquid content which would run off of a plate.

A large plate intended for serving food may be referred to as a platter.

@aaronasterling gives a good explanation of what they mean as pieces of crockery. They can also both mean — drat, I can’t find a good synonym now! — let’s say, menu items you might order at a restaurant:

My favourite dish at the Murray Avenue Grill is their spinach salad with candied pecans.

or:

The Ketchery has the best seafood plate in town.

These arose in obvious metonymy from the crockery usage, but have developed specific connotations. Dish in this sense is much more general, and used (afaik) across most or all of the English-speaking world. Plate is less usual, and more associated (I think) with North American usage, and with fancier restaurant subcultures.