In the last three to five years, this answer has changed quite a bit. It used to be dLSR vs. small-sensored point-and-shoots but today, there are a number of different cameras that fill in this very large gap, and the answer for this may still be evolving, as manufacturers continue to experiment with ideal format sizes and camera feature configurations.
Currently, the main "categories" that I see of cameras that offer you more options than a 1/2.3" format pocketable P&S camera are:
These cameras have dSLR-style bodies, but are still fixed lens, and for the most part are still 1/2.3" format (5.6x crop), but may offer a flash hotshoe, RAW capability, and full manual mode, and are nearly always going to have a superzoom lens on them. They tend to be weak for low light photography, but pretty good for daylight photography, macro, and general walkaround shooting. The Panasonic FZ, or Canon SX lines are examples of bridge cameras.
These cameras typically offer a slightly larger 1/1.7"-format sensor (~4.5x crop), and make the tradeoff of reach for low light capability. They generally sport shorter and smaller zoom lenses, but with larger maximum apertures (f/1.8-f/2.8) that can let in more light. Most offer RAW capability and full manual mode, and some may sport a flash hotshoe. The Canon S and G lines, Panasonic's LX, and Olympus's XZ lines are examples of enthusiast compacts.
Possibly a sub-category of enthusiast compacts, these cameras can often be as expensive as dSLR kits. They are fixed-lens, like most compact cameras, but have much larger sensors in them. The range of sizes, however, varies from 2/3"-format (4x crop) to full frame (1x crop), and just about everything in-between, including APS-C. The features will also vary widely, but they all offer RAW capability and full Manual mode. Examples of this class of camera include the Fuji X20, Fuji X100, Sony RX100, Sony RX1, Canon G1X, Nikon Coolpix A, etc. This is where there seems to be the most experimentation going on with camera companies trying to find "the winning formula" and where the picture is still likely to change.
Mirrorless Interchangeable-Lens Cameras
Also known as MILC, EVIL (Electronic Viewfinder Interchangeable Lens) and CSC (compact system camera). These cameras, like dSLRs, have interchangeable lenses and larger sensors, but work more like compact digital cameras with the light coming through the lens directly to the image sensor (no mirrorbox). The image sensor is also used for exposure evaluation and autofocusing, unlike a dSLR that uses two separate sensor arrays for those purposes. As a result, these cameras are typically smaller and lighter than dSLRs. The sensor formats, again, range across sizes from 1/1.7" (Pentax Q) to 1" (Nikon 1; 2.7x rop), 4/3" (micro four-thirds, EOS M; 2x crop), APS-C (Sony NEX, Fuji XF; 1.5x crop), to full frame (Sony A7).
Because mirrorless cameras are system cameras, they can be astronomically more expensive than the other types, since you'll typically be purchasing other bits of the system (lenses, flashes, etc.) as well as the camera body. And because these systems are so much newer than dSLRs, they can't leverage a back catalog of film-era lenses and 3rd party support, like dSLRs can. But many of the mirrorless cameras perform so well for the size/weight reduction that a number of shooters are moving to mirrorless from dSLRs. It's probably best to consider these systems as an alternative to a dSLR, rather than a bridging step to one.