Alright, let me break it down, real nice and simple here:
DSLR cameras
Pros:
- high quality images
- variety of interchangeable lenses for different perspectives and different types of photography requirements. From super-wideangle, to wide-angle, to normal, to short-tele and then long-telephoto, it all depends on what lens you use.
- more rugged bodies than P&S's and Bridge Cameras, some are even weatherproof (my D200 can take pics in drizzle or sandy/dusty conditions, I've shot on the beach with sand all over it before. All I had to do was just blow/wipe off the sand. In the drizzling rain. With a wet beach towel.)
Look what Joe McNally did to a Nikon D2x, that Nikon Corp. LENT to him to shoot the Baja rally in 2005:
- Takes images in "RAW" format (different from jpeg, png, etc) - RAW has more digital info so when you edit in PS, you have much more leeway when editing and lose less data when converting to jpeg
- faster startup and shooting times than a P&S/Bridge (the D40 will startup and fire off a shot in 0.45secs, while a Canon S5is Bridge camera will do that in 1.6 seconds
Cons:
- Expensive to buy and repair if damaged (can reach into the thousands, if you really mess things up)
- Not all lenses are supported with basic level DSLR's (the D40/D60/D3000/D5000 can't take older lenses made before 2001, due to lack of built-in motor in the camera to focus the lenses. There is however, a positive paradox with the older legacy lenses and the D40/D3000/D5000 bodies, but this is a slightly technical tidbit)
- Accessories can be expensive (lens, bag, cleaning kit, filters, flashes, tripods, softboxes, lighting umbrellas, editing software.... and so on).
- Learning curve involved (mitigated by a HUGE data-pool on the internet - Flickr/photo.net/digitalphotographyschool and other websites)
- Bodies and lenses are bulky, may be difficult to travel or walk around with all day if you have a bulky setup
Bridge Cameras
Pros:
- Can take both pictures and video (most of them can)
- Lenses are built in, and most can do superzoom, up to 18x or even 20X zoom (good for spyshots or if you're in a football/cricket match)
- More compact than DSLR's, so more suitable for all-day walking around. I used to have my Canon S3is around my neck all day long with little discomfort. Not so with the D200 and battery grip, that thing weighs a ton.... or feels like it after 3-4hours of walking with it.
- Cheaper than DSLR's, and some even come with interchangeble lenses now (Olympus does this, I think)
- Shoots faster than a normal P&S camera, some even do 1.5 frames per second in burst-mode.
Cons:
- Still a bit more expensive than regular P&S's; a Bridge Camera typically starts at around $400 USD while a good quality P&S can start at around $175-$300 USD
- More bulky than P&S's, so won't readily fit into a shirt pocket; more like a purse or backpack
- Lens usually is not interchangeable (with a few exceptions like Olympus), and due to engineering limitations the glass isn't the sharpest at the more extreme zoom focal lenghts; and can be prone to optical artifacts (it's hard to engineer a perfect lens that can zoom ALL the way to 20X without compromising a few features).
Point & Shoot (P&S)
Pros:
- small and compact, perfect for shirt-pockets or carrying in a small case on the pants-belt or around the neck
- large variety to choose from, different companies
- cheaper than DSLR's and Bridge Cameras
- Often easier to use than DSLR's and BC's, hence the name "P&S". Just select "Auto" and press the shutter release - voila, instant snapshot
Cons:
- image quality usually can't touch a DSLR with even a basic kit lens
- smaller image sensor means not much depth of field, so you don't get the background-blurring-effect (called "bokeh" by DSLR fanatics) that makes a nice, pro-level photo
- smaller bodies means more delicate electronics. Drop a P&S and say bye-bye to it, time to buy a next one. Shoot with one on the beach and get sand on it, and it's bye-bye mister lens-motor, the damned thing will stick (happened to me)
- slow shooting speed. How many times have you tried to get a quick shot, only to hold down the shutter button, and then wait 1-2 seconds for the camera to focus, then beep, and then take the shot, and then another 1-2 seconds for the image to come up for review, and then 1-2 seconds to get ready again for a next shot? The Nikon D300s can be turned on and fire off 4-5 shots in burst mode in under 1.5 seconds.
- Difficult to get good shots in low-light conditions. This is because of a small sensor, and an even smaller lens. Forget what the marketers tell you about a P&S having "f/2.8, which means a very wide aperture". The size of the hole in these lenses is still very small (compared to a DSLR lens), so not much light can enter in dim conditions. Result = blurry or grainy shots. Using the on-camera flash doesn't help much, you end up with the crappy, overblown snapshots with red-eye or the "white-out nightclub look".
- Not as ergonomic as DSLR's or even Bridge Cameras. Those were designed to meld into the shape of your hand; not so with the rectangular-shaped P&S's. Thus harder to hold these effectively, without resorting to the "thumb&forefinger" grip, with the other 4 fingers of each hand splayed out as if making the "OK" sign while holding the camera.
This can get uncomfortable after a couple hours. Better to hold your P&S like this:
This gives you maximum stability and sharper pics, less blur from camera-shaking.
Hope this helps clear things up, and helps you make an educated decision on what to get.
.... trust me. It had times i literally sank into a depressive state because some jack@$$ outbid me on a lens on Ebay. 
