Most bridge players understand roughly what to expect from a "natural" bid, but it's quite a difficult term to define precisely. This is unfortunate because, as well as being a useful word for describing what a bid means, it's also often used in system regulations and alerting regulations, where you really need a good definition.
The meaning also varies slightly depending on where you live. Let's look just at suit bids. In America, most people seem to consider a suit bid to be natural if it promises length in that suit, even if there is some additional information given. For example, a Muiderberg 2S opening (showing 5 spades and 4+ cards in a minor) would be described as natural. They would say it is "natural but conventional". The WBF, on the other hand, defines "natural" to be the opposite of "conventional". Of course, "conventional" is another term which is extremely difficult to define, but it is clear that the Muiderberg 2S is conventional, and so is not natural according to the WBF's definition.
In England our understanding is somewhere in between. The old Orange Book (pre-2006) summed it up quite well: a natural suit bid was
a bid of a suit which shows that suit and says nothing about any other suit. ...
So a Muiderberg 2S opening is not natural, because it shows length in another suit (even though that suit is unspecified). But, unlike with the WBF's definition, it is still possible for a natural bid to be conventional. We would say that the opposite of natural is artificial. An example of a bid which is conventional but not artificial is a 1D opening which promises an unbalanced hand. This certainly shows diamonds, and says nothing about any other suit, but there is an additional agreement about the hand as a whole which makes it conventional.
Because I learnt my bridge in England, this is the definition which feels most right to me (and not just because I'm an avid reader of Orange Books! - this really is what people mean by "natural" in England, give or take some of the details). Unfortunately, both of the conditions in that old OB definition are slightly faulty.
First of all, what does it mean to "show that suit"? The old OB definition continued:
The suit shown will be at least four cards before opener rebids but may be three cards from then on; exceptionally a bid of 2C in a 3=4=3=3 hand precisely in response to 1S is considered natural.
This definition appears to make bids such as 1H (pass) 2H artificial if they could be made on three cards. In practice, this oversight was just ignored - no-one was suggesting that these bids should be alerted - but it did look a bit silly. It was corrected in the new version of the OB, where the definition of a natural suit bid is:
A bid of a suit which shows that suit and does not show any other suit; the suit shown will be at least three cards long except that preference bids and raises may be on shorter suits. Note that in earlier rounds of bidding a natural suit bid usually shows at least four cards.
This is a much better definition of what it means to "show" the suit bid. It's also an improvement for the bids which show only three cards suit. According to the old OB defintion, a 3-card 1H opening was alertable because it was not considered to be "natural". But this doesn't seem right - it surely is a natural bid, but ought to be alerted because it is unexpected.
The second part of the definition is much harder to get right, and neither of the two EBU versions really works properly. According to the old regulation, a natural bid should "say nothing about any other suit". Presumably this was discarded when it was realised that nearly every natural bid shows something about the other suits, even if this is only from the negative inference that some other natural bid could have been made instead. And, for example, a pre-emptive opening bid which is played as denying a side 4-card major should still be considered to be natural. But the new version goes too far in the other direction, so that (for example) a 3H bid which shows heart length and club shortage is considered natural. This isn't a problem for the alerting regulations because club shortage is still unexpected, and therefore alertable, but it doesn't correspond to what we really think of as being "natural". I think the old definition is closer to the truth here, and they just needed to make an exception so that a bid could still be natural if it denied a certain amount of length in another suit (or suits).
Still, there isn't really any obvious way to define it perfectly, and it's interesting to see how various different authorities try (and fail) to do it. And that's without even considering what a "natural" no-trump bid should mean.