Some quick sketching...
People often talk about the whole being larger than the sum of the parts.
The truth of this is commonly seen as showing that reductionism is wrong, as something new has emerged. This is actually a radical, extreme version of emergence, known as strong emergence.
There’s no evidence that strong emergence is true, and it’s not a very ‘respectable’ position. The whole isn’t really greater than the sum of the parts, because it isn’t just the sum of its parts: it’s a result of the combination of the parts in a particular configuration.
There isn’t anything genuinely new that emerges. It’s just a matter of the configuration or interaction of the parts. This is known as weak emergence, and seems to be really what people talking about emergence are talking about.
At other times, when people are talking about emergence, they’re not talking about the entity itself, but the effects of it. For example, saying that a fantastic painting is far greater than just the canvas and the paint it’s made of. This is making the mistake of overlooking the particular configuration of the paint, but it is also that when they’re talking about the painting they’re really talking about the effect when you are perceiving it.
What I think underlies these mistakes, and what doesn’t tend to be brought up, is that when people are surprised by a whole -- when they are likely to bring up the saw about the whole being greater than the sum of the parts -- they haven’t actually really calculated what the sum of the parts would be. If they had, they’d have known what the whole is. What is really going on is that the whole is greater than their intuitive conception of what the parts would add up to.
(This reminds me of the criticism Dennett makes, in Consciousness Explained, I think, of things like Searle's Chinese Room argument -- and some other philosophical thought experiments, I think -- that they invite you to imagine something, and draw conclusions based on your imaginings, but which you can't actually concretely imagine, but rather just have a vague impression from).