It's always interesting to see how Wikipedia is handling philosophical topics. The current Wikipedia discussion on Hume's criticism of the design argument foregoes a summary of the criticism itself (which is probably wise, as there is no general consensus on the whole shape of Hume's criticism) and instead just lists a handful of points made:
Here are some of his points:
1. For the design argument to be feasible, it must be true that order and purpose are observed only when they result from design. But order is observed regularly, resulting from presumably mindless processes like snowflake or crystal generation. Design accounts for only a tiny part of our experience with order and "purpose".
2. Furthermore, the design argument is based on an incomplete analogy: because of our experience with objects, we can recognise human-designed ones, comparing for example a pile of stones and a brick wall. But to point to a designed Universe, we would need to have an experience of a range of different universes. As we only experience one, the analogy cannot be applied. We must ask therefore if it is right to compare the world to a machine — as in Paley's watchmaker argument — when perhaps it would be better described as a giant inert animal.
3. Even if the design argument is completely successful, it could not (in and of itself) establish a robust theism; one could easily reach the conclusion that the universe's configuration is the result of some morally ambiguous, possibly unintelligent agent or agents whose method bears only a remote similarity to human design. In this way it could be asked if the designer was God, or further still, who designed the designer?
4. If a well-ordered natural world requires a special designer, then God's mind (being so well-ordered) also requires a special designer. And then this designer would likewise need a designer, and so on ad infinitum. We could respond by resting content with an inexplicably self-ordered divine mind but then why not rest content with an inexplicably self-ordered natural world?
5. Often, what appears to be purpose, where it looks like object X has feature F in order to secure outcome O, is better explained by a filtering process: that is, object X wouldn't be around did it not possess feature F, and outcome O is only interesting to us as a human projection of goals onto nature. This mechanical explanation of teleology anticipated natural selection. (see also Anthropic principle)
6. The design argument does not explain pain, suffering, and natural disasters. See Problem of evil.
(6) is right, although it is misleading simply to lay it out there without development; Hume is quite clear that the design argument's failure to explain pain, suffering, and natural disasters is not a problem for the design argument itself but for a particular use of it. And Hume definitely does make point (5), which can indeed be seen as a sort of loose anticipation of natural selection if we are thinking only of biological organisms (Hume himself, of course, is thinking more generally than that). Point (3) as laid out here seems to me to be confused and to muddle more than one thing together, but it does identify an important and often overlooked point (related, actually, to point (6)), namely, that a hefty portion of Philo's criticism in the Dialogues is devoted not to attacking the design argument but to attacking Cleanthes's claim that the design argument is sufficient for all religious purposes. The first sentence in (1) is certainly false in Humean terms, and I think Hume makes clear enough from the discussion that he actually rejects it. For exactly the same reason, the "analogy cannot be applied" point in (2) is not right: Philo's point is precisely that the analogy can be applied -- and so can a very wide variety of other analogies. This is actually important: Hume does not reject the analogy, which he has Philo explicitly affirm more than once in the Dialogues. Nor does he deny that you can make some sort of inference on the basis of it. Philo instead argues that the analogy, and the related analogical inference, can't do what Cleanthes thinks it can.
All in all, though, it's a decent attempt; I've come across professional philosophers making worse and more obvious errors in discussing this topic. Like I said, I think it was a wise move to focus on 'points' rather than to try to lay out the overall shape of the argument, and most of my disagreements would be on matters of wording and organization, both of which could be better. The only major error is the one about Hume's view about the feasibility of the analogy, and it's an easy one to make given the complexity of the discussion. And whoever wrote it caught on to the importance of the relation between the design argument and religion, which is definitely an important part of the argument, but easily missed. I was pleasantly surprised with this Wikipedia discussion: it's a very hard topic, involving a great many controversies, but was handled reasonably well here. I hope future revisions keep the same basic approach and just refine the listing of points (ideally with indications of which Parts in the Dialogue make these points).