Pedal cleats would normally come with the pedals but if you have managed to wear them out you can buy them separately.
There are a few different types of fit and there are lots of brands that offer a full range of cleats to fit pedals from other brands.
Shimano were the first company to use a pedal and cleat combo and the term SPD actually stands for Shimano Pedalling Dynamics. The SH51 cleat from Shimano is still the main shape and style for any mountain bike pedal. It is made of metal and is around the size of a 50p coin. This cleat is fixed user 2 countersunk screws that screw into a plate within your shoe. Over the last few years some shoe brands including Shimano have the bolt option on lower end road shoes too.
Road pedals and cleats have a few more options. The cleat is usually larger than the mountain bike option (close to the size of your palm) and they are made of plastic. These road cleats are triangular in size as most have a 3 bolt fixing. The bolts fit within the cut outs of the cleat. The most common fit for road pedals are Shimano SPD-SL, Look Keo, Look Delta and Speedplay. The Speedplay cleat are the only rectangular offering and are made up of a couple of a layer of plastic topped with a metal spring and cleat. To fit these onto a 3 bolt shoe you must use a cleat converter.
The sizes and shape of all cleats are based on the brands pedal designs. Road cleats are larger help to keep your foot firm in the pedal under great pressures. They are commonly available in 3 options. This is so the rider can allow their ankles more float/movement for comfort and pedalling style. Mountain bike cleats are metal and smaller so the shoe can still be walked on and so that rocks and stones won't destroy the cleat whilst walking.
Cleat covers are available for all cleats to protect them whilst walking on them however most only use them on road style cleats as the metal mountain bike cleats being metal can take a fair bit of walking before there are signs of wear. The covers are rubber so they are easy to clip and store nicely in a pocket or saddle bag.