I'll try to give an intuitive perspective on the significance of the specification in interpreting the claims.
The claims cover what one of ordinary skill in the relevant technologies would understand they cover. If this hypothetical person of ordinary skill had read the entire specification prior to reading the claims, this person would have some understanding of what the various terms of the claims mean. This understanding can come from dictionary meanings, from terms of art, and from context and/or explicit definition in the specification.
I personally get some insight into this process when I read an application for computer technology written by someone with little or no experience in that technology. Terms are used in odd ways, and I have to intuit what the terms are supposed to mean in that application. One that comes to mind is "primary storage" and "secondary storage". Historically, those meant volatile memory (e.g., RAM) and non-volatile memory (e.g., disks), respectively. But, when you read the spec, it's pretty clear they meant something like high-speed (or in-processor) cache and system memory, respectively. Sometimes they just mean two different collections of storage of equal construction just used for different purposes. When the spec and conventional interpretations disagree, which controls? My sense is that it's a "you know what we mean" standard.
The law was moving toward dictionary-only interpretations. In my opinion, it was part of a trend of eviscerating patents of any value for any modest mistakes -- a host of gotchas like Festo. In Philips (noted by Isaac), the court moved away from that and toward a more reasonable standard of allowing the spec to shed light on what various terms mean -- getting closer to my intuitive perspective above.
Now, when writing an application, I always do my best to avoid any contradiction between conventional definitions and my usage in the application. If you're looking at an application after-the-fact, things are in your favor if you're following a common-sense, "you know what I mean" interpretation of a term and you're fighting an uphill battle if you're striving for a more strained, formalistic interpretation of a term.
Regards.