Here's a screenshot of how I run my Aperture. Click through to Flickr for notes.
Firstly, I have an @Review folder at the top level. This is where new projects go. When I insert a card, I just point Aperture's import sheet at the @Review folder and it makes a new project under there for me. The point of the @Review folder is that it contains all the projects that I haven't finished rating (at least up to three-stars) and editing.
Next, I have one folder per year and within each one folder per month. I use the month number in the folder name (for example, "01 - January") so that they sort in chronological order. Aperture uses the alphabetical ordering of items in the project list so, without the numbers, August would come top of the yearly folder. Thankfully, Aperture does use natural sorting for numbers!
I find this Year/Month arrangement to be pretty useful. I find it much easier to remember things by date than by 'theme'. Theme-based organisation is too fuzzy: I doubt I would be as consistent in my organisation, and I would be constantly wondering about whether I should reorganise. In any case, keywords and smart albums are a better way to overlay theme-based organistion on top of date-based organisation.
When it comes to finding projects, this structure is pretty helpful. If I don't remember exactly which month it was, I can likely at least remember which year it was. Then, I can click one of the year folders and all the images for that year show up. A quick scan then shows me if I was right. I also like this structure, because it makes your Aperture library start to become a kind of informal photographic diary, which I love to look back through.
What is a Project?
I see a few Aperture users getting hung up on when they should create a project as opposed to albums within a project. I like to keep my projects pretty narrowly focused and have quite a few of them. Because I'm organising by /Year/Month/Project I usually end up with, at most, about 15 projects in a month.
A project corresponds to one "shoot", however that's defined for an amateur photographer. In most cases "one project" == "one day out taking pictures". In some cases, though, I'll split a day's shooting into two projects if I did two separate things. A good example of that would be if I went to shoot two different places in one day, I would make a project for each place.
I don't have many long-running projects. Most are complete within a day but, in this organisation strategy, it only matters if I have projects longer than one month, which I almost never do. If I do have projects that run longer (and I do want to work on some of these), I'll put them at the same level as the monthly folders, in the year folder.
If I take a long trip, say the ten-day trip around WWDC last year, I'll usually put all the images in one project and then break the project up using albums within the project. I have in the past gathered related projects in folders within the month, but I prefer to be able to see all the images in a trip just by selecting the overall project.
Another advantage to keeping projects small and tightly focused is that the project is Aperture's basic unit of import and export. I use project export a lot to move images from my laptop to my desktop machine. I recently helped a user in the Flickr Aperture group who had imported all his images into one project and organised them by albums within the project. His problem was that he needed to export some images to another Aperture installation, so had to split up his one mega-project into smaller, more focused projects.
Brown Folders and Blue Folders
Aperture has two kinds of folders and, like many things in Aperture causes no end of confusion until you understand the rules for them. The simplest rule I can express is this: Blue folders create structure outside a project and brown folders create structure inside a project. You can't drag a blue folder into a project and you can't drag a brown folder out of a project.
If you have a project selected and choose File > New Folder, you'll get a brown folder inside that project. If you have, say, a blue folder selected and choose New Folder, you'll get a blue folder inside that folder.
In my organisation scheme, I don't use brown folders at all. This is mostly due to the fact that I keep my projects small and focused. They don't have a lot of structure inside them. If I did larger projects with lots of variants and possible destinations for the images, then I would probably do more organisation inside each project.
One reason I like to structure with blue folders is that they show their aggregate contents in the thumbnail browser, whilst brown folders don't (and I don't know why not). This means that I can click any year or month folder and see all the images in that year or month in thumbnail view. Unfortunately, you can't select a brown folder and see all the images within that folder.
Further Reading and Listening
Here are some other resources that I've found pretty useful when understanding Aperture's project organisation philosophy:
Aperture product manager Joe Schorr on the O'Reilly Media Inside Aperture Podcast:
The Bagelturf blog is full of good stuff on working with Aperture.