Pomes are a type of fruit. It is a purely botanical classification however. No one outside of botanists and French-speakers is likely to have heard of them. ("Apple" is "pomme" in French.)
The term itself is actually derived from the Old French word for apple. This is shown in its meaning which is kind of just "something like an apple." The simplest way that I can put it is that a pome is a fruit where the ovary is surrounded by a fleshy tissue called the hypanthium made from the bases of the flower petals. This makes the pome a type of accessory fruit.
As you can suppose, the apple is by far the most common example of a pome. The pear is the only other pome common in America.