Not just in biology!
From the dictionary: Emergent properties - arising and existing only as a phenomenon of independent parts working together, and not predictable on the basis of their properties.
In other words, it's where a collection of things working together have new properties that are more than just the sum of the parts. For example, organisms are more than just bags of cells. They do things you would hardly expect (or predict) from the properties of mere cells, such as howling at the moon, being allergic to peanut butter, filling the atmosphere with oxygen and our fuel tanks with oil. These are emergent properties.
Personally, I think emergence is an illusion. Who is to say what is predictable? The fact that the same types of systems reliably produce the same kinds of emergent properties indicates they are quite predictable on some level even though it is not yet understood by scientists. Come back in a decade or two.