Until recently, digital cameras came in just two flavours: big "Single Lens Reflex" (SLR) cameras, and everything else (called "compact" cameras). The easy way to tell the difference is that you can swap lenses on SLR cameras, while compact cameras have their lens permanently stuck on. Then came "mirrorless" cameras to shake it all up (see below).
SLRs are bigger, heavier, more expensive, make a noisy "clunk", can get dust all over your pictures, and (with all but the latest models) you can't shoot video, and you have to look through the viewfinder to frame up pictures. And then there's all the hassle of swapping lenses.
So why would anyone buy them? Because they are better in every possible photographic respect.

A digital compact camera (left)
and a digital SLR (right)
On our photography courses, I meet lots of people who are thinking about getting a new camera, and want to know if SLR cameras are worth the extra money and bulk. This page is for them. I’ll show the key differences in the types of photos you can take with each type of camera, to help you decide what fits your needs best.
Do you have a need for speed?
SLRs shoot more quickly and don't leave you waiting – using an SLR for the first time is like jumping into a sports car after using a bicycle. If you want to photograph sports, you need an SLR. They focus faster, take pictures the moment you press the button, and can take lots of photos in a row without pausing for breath.
Here’s a 1-second sequence showing how well an SLR can track a moving target – Isabella is running towards me at full tilt indoors.

SLRs focus and shoot faster than other cameras, so you can capture action
Compact cameras simply cannot focus and shoot this quickly. This sequence comes from expensive gear: a $1,500 Canon EOS 40D SLR with a $3,000 lens courtesy of Ted's Cameras in Brisbane city. It got half of the running shots in good focus indoors. With a bit of practice, even a cheap SLR will give you some shots in focus, particularly outside. With a compact camera, you’ll be lucky to get a sharp shot at all in these conditions.
How dark can you go?

SLRs give much better pictures than compact cameras when the light is dim. Even the best compact cameras struggle indoors without flash, giving blurry or speckly pictures. SLRs – even the cheapest ones – do much better. That’s because SLRs have bigger sensors - 5 to 10 times bigger - so they need much less light to give smooth, sharp pictures.
The photo on the left was taken in dim lighting indoors while Carmen watched Play School. I took photos of her with different cameras to show the differences in quality. I've enlarged the eyes from each photo below. I used a cheap compact camera, a good compact camera, and SLRs with different lenses, all taken without flash.

SLRs give much better quality photos indoors without the flash than compacts can
SLRs let you blur the background
SLRs let you make the background totally out of focus if you want to, making your subject stand out. Compact cameras with small zoom lenses just can't do this. Compact cameras with long zooms can blur the background a fair bit, but nowhere near as much as an SLR can. These shots tell the story:

SLRs can still get everything in focus too if you want to.
Change the lens, change the camera
SLRs can use different lenses that let you to take photos that you can't with a general-purpose lens. You take one lens off and pop on a different one, turning your camera into a specialised tool for the thing you're photographing. You might already have some SLR lenses from a film SLR camera - this page tells you if they will work on a modern digital SLR.
Here are some of the things that you can do with specialised lenses.
Some lenses let you shoot in dark conditions without flash. This shows off the natural light, and gives your shots a journalistic look. You can get shots like these with a tiny $150 lens called a 50mm f/1.8. It doesn't zoom, but it lets you take sharp photos almost anywhere - even in this dim hospital delivery room
Some lenses help you exaggerate perspective for dramatic landscapes. See the way the shadow of the tree seems to zoom away from you? Compact cameras can't do this to the same degree. You can get shots like this on an $800 lens called a super-wide-angle zoom (a 10-20mm zoom).
Without a special lens, SLRs can't focus as close as most compact cameras can. But if you want to get really close, like the shot of this spider, a compact won't do it. You need an SLR and a "macro" lens. The lens was almost touching the spider. Macro lenses start around $400
Other lenses let you zoom in super-close for sports or wildlife. They're physically big lenses, and range from $300 to "You must be joking!". Some compact cameras can zoom in a long way too, but they don't focus and shoot fast enough for sports, and they won't get the background as blurry.
You don't have to get such specialised lenses. Some lenses will do several of these, but they tend to be either huge and cost thousands, or give pictures that aren’t as good for making big enlargements.
SLRs can accessorise...
SLRs can take different types of bolt-on flash, giving you more flexibility for shooting indoors. Why would you want a bolt-on flash when most cameras have one already built in? Four reasons. Bolt-on flashes can keep up with rapid shooting, so they don't leave you waiting ages for the next shot. They're more powerful, so you can shoot from much further away or light-up a whole room. They cause much less red-eye, and most bolt-on flashes also let you do this:
Both of these photos were taken with flash, but the photo on the left has the flash pointing up at the ceiling, so everyone is evenly lit. In the photo on the right, the flash is pointing forwards (as do all built-in flashes), giving harsh shadows and a dark room.
Some let you shoot with one or more flashes off the camera for a real studio look.
SLR lenses take filters, which are particularly handy for creative landscape shots. Very few compact cameras allow you to use filters.
SLRs forgive mistakes
Because photos from SLRs start out smoother with less speckling, you have more wiggle room to improve your photos after you've taken them - either on your computer, or when you print them at a kiosk. You can make a photo brighter, or enlarge just part of it without the photo getting as blotchy or mushy as it would with a compact.
SLRs are much more forgiving of mistakes when you take photos; with a compact, you need to get it closer to perfect on the spot.
Summing it up
So if you're after quality, speed, creativity or low-light shooting and you don't mind a lighter wallet and a heavier camera, an SLR is the way to go. If convenience is more important, go for a compact camera instead.
I’m meeting a growing number of people who are disappointed with their second digital camera when they’ve chosen an expensive compact. The camera is fine, but they believed the marketing hype that it will be as good as an SLR. It's not.
The best of all worlds may be to get one of each: a cheap compact AND a cheap SLR, rather than a super-expensive camera that's trying to be all things at once. If you shop around, you can get both for under $1100 in total. You'll have a go-anywhere compact with you at all times, and a good-quality SLR for more demanding or creative shots.
Is this the end of the SLR?
There's a new type of 'Mirrorless' camera that promises a perfect mix of SLR quality, changeable lenses and compact camera size. I expect it'll be the start of the end of the SLR. Panasonic launched the first - the Lumix DMC-G1 - in 2008, and there have been over 30 new models since from every major manufacturer. These new 'Mirrorless' cameras made up a massive 40% of camera sales in Japan in 2010, and they became even more tempting in 2011 as manufacturers refined their early models and finally fixed their Achille's heels: price and autofocus speed. Now in 2012, they're looking like strong candidates for your next camera. Mirrorless cameras completely scramble the neat distinction between compact cameras and SLRs. Pentax's Q behaves like a compact in that it can't blur the background well; Nikon's J1 allows a touch more blur, Panasonic's GF3 a bit more, while Sony's NEX series rubs shoulders with SLRs for blur. In just three years, photographers have gone from having two types of brush for painting our pictures (compact or SLR), to having five... with lots more on the way. It's an exciting time to be taking photos!
What's in a name?
“Single Lens Reflex” is a phrase that doesn't mean much these days, and describes the internal workings of the camera. It used to mean a lot when there were “Twin Lens Reflex” cameras with (you've guessed it!) two lenses – one acted as the viewfinder and the other took the picture.
The “Reflex” part refers to a mirror that sits inside the camera between the lens and the sensor, and diverts the picture through the viewfinder until the moment you press the button. Then it flips out of the way (making the camera's “clunk”) to let the light through to the sensor to record the picture, before flipping back again. It was a big innovation in the 1950s, allowing photographers for the first time to see a picture the right-way-up through the lens that would take the picture. Nowadays, every digital camera lets you do that without needing a mirror – you see the picture on the LCD screen on the back. But the name is with us for good.
The flipping mirror was needed in the days of film, but it's beginning to look like a kludge in digital days. Panasonic ditched the mirror in their revolutionary G1, and it's how they made the camera and lenses smaller. The writing's on the wall for the mirror - once it's gone, all digital cameras can be smaller, quieter, faster machines that can shoot movies too. I can't wait!