Manure is organic material that is used as a fertilizer. Manure can be from animals, plants ("green manure"), or compost.
Most often when people say "manure" they mean animal feces, possibly mixed with bedding -- straw, hay, or wood shavings - from where the animal is housed. Different animals produce manures with different nutrient contents, but most are a good source of nitrogen.
"Green manure" is plant material used specifically for soil improvement. Certain crops, such as legumes (clover, beans, etc.) can fix nitrogen from the air, so planting these adds nitrogen to the soil. Other crops, such as winter rye, will scavenge nitrogen left over from a previous crop; this protects this critical nutrient from being lost through leaching. When the green manure crops are tilled in, the nutrients contained in the green manure benefit the following crop.
compost is simply decomposed organic material -- including both of the above.