You're looking for balance, so the answer is double edged:
- Stay compelling by avoiding drifting off on tangents; don't explain about your world where it isn't relevant to the story at hand and to the readers' interest.
- Convey your world by choosing a structure and a plot where the elements you want to convey feature heavily. If your setting elements are crucial to the plot, then the setting itself will be compelling.
It sounds as though you're pretty clear on the necessity of not boring the reader with mere detail. I've often heard this referred to as "infodumping"; I find the term very appropriate: what you want to avoid is not information or exposition - it's presenting information or exposition in an uninteresting, non-compelling manner; it's when the reader feels like the information is being dumped upon him. So this part of the answer is simple: do not present information without a clear idea of why the scene is immediately interesting to the reader (or how you can make it so).
The second part of the answer is more subtle, and it works in the opposite direction. We don't want to provide any unnecessary information; that doesn't mean we need to discard all the non-crucial information - instead, we can make it crucial.
What you do here is identify the setting elements that you most want to convey. Then you construct the story around those elements - so that those places, characters, customs, linguistic oddities, etc. are the building blocks of your plot; the interaction between them is the central focus of the story. Don't be satisfied with simply involving these elements - setting the story in Location X will not be very effective if the story could just as easily have been set in Locations Y or Z with no real difference to the story. A character grieving over his wife might be perfectly interchangeable an engineering student who hates his advisor, if the story is about a protagonist running away from a mutant zombie. Instead, you're looking for situations and interactions that rely on the unique characteristics of your setting elements. If you can manage that, then the solution to your question falls into place - the readers will want to understand your setting, and the elements you've chosen, because they're what the story is all about.
- Choose the elements you want to showcase;
- Construct the plot (or subplot, or flavor - anything sufficiently compelling) around these elements;
- If any of your major elements could easily be swapped for something else, then you haven't involved it deeply or uniquely enough.
You've still got a lot of work cut out for you - in order to pull off a plot relying on your unique setting details, you'll need to manage compelling exposition so the audience will understand what's going on - and not need to sit through a lecture before the action starts. But that's a lot easier to do once the details are intertwined with the plot, rather than being entirely incidental to it.
A great example of a writer who does this well is George R.R. Martin in his Song of Ice and Fire series; he's certainly constructed a complex and involved world, but he:
- Shows it off little by little
- Creates a wide cast of characters, each exploring different parts of the world
- Always (well, almost always) makes the exposition immediately relevant and compelling to the plot at hand (for example, in "Hedge Knight," he describes a huge political situation and family conflict - by making the protagonist's goals dependent on watching the larger picture, and interacting with different aspects of it)
Similarly, often mystery stories are great for exposing setting - because they give the reader incentive to be interested in minute details about the setting. These details can be crucial to solving the mystery, so the reader's happy to go into investigative mode!
A last related point - Orson Scott Card refers to "the mileau story" - a story whose entire structure is a journey through the setting; things may happen along the way, but the journey itself, and the sights seen along the way, are the focus. This certainly is a structure that might be well-suited to conveying a setting, though I often feel this structure makes it hard to plot compellingly - a journey through a bunch of disjointed and practically non-interacting locations is hard to get across as a coherent, compelling whole. As examples, I'd point to Neil Gaiman's Stardust and Neverwhere; to Gulliver's Travels and Around the World in 80 Days; to The Phantom Tollboth, and probably even to Lord of the Rings itself.
Hope this is helpful :)