There are a lot of good reasons to choose a camera over a consumer camcorder for shooting video, including larger sensors, which tend to deliver better tonal range and enhanced depth-of-field flexibility, and better photo quality. And an interchangeable-lens camera (ILC) -- a dSLR, fixed mirror (Sony's SLT series), or mirrorless model -- imparts huge creative and logistical benefits over a typical point-and-shoot design.
Most consumer camcorders have pretty ugly polygonal bokeh until you hit the $1,000 or so price point. With an ILC, even the cheap lenses deliver better handling of out-of-focus areas.
So why do camcorders still exist? For one thing, they're physically optimized to be held for longer periods of time without inducing fatigue. Holding a camera with the LCD at eye level or the bulky body with an LCD tilted or flipped out can be awkward. Cameras also tend to be a mixed bag on the autofocus (AF) front. Mirrorless designs -- point-and-shoots and ILCs -- tend to have an advantage because they're built around contrast-based AF systems that are more suited to continual operation than the phase-detection AF systems integral to dSLRs. (Here's a discussion of the two technologies.) The videographers who first used dSLRs accepted the need to manually focus -- and to retrofit their cameras with the lens add-ons necessary for smooth, comfortable focus operation (a follow focus) and LCD magnifiers called loupes for precise operation on the relatively small, low-resolution displays. But not everyone wants to surround their nicely sized camera with an unwieldy Erector Set-like rig. So if you just want the video equivalent of a point-and-shoot, my recommendation is you stick with a mirrorless ILC.
Some key features to consider while deciding on a model include manual controls -- the ability to adjust shutter speed, aperture, and gain (ISO sensitivity) -- support for an external microphone or headphones, and a tilting or articulated LCD. While many cameras have manual controls, watch out for the caveats. For example, the Sony Alpha SLT-A77V has nice autofocus, but you can use it only in automatic mode. Also, some cameras only let you adjust aperture, not shutter speed; if you can't control the shutter speed, you can frequently end up with a jittery, unpleasant look to video shot in bright light. While focus peaking, which uses edge highlighting to show the scene moving in and out of focus, is a huge advantage for manual focus, at this point only a few cameras seem to have incorporated it.
Also, keep in mind that you don't necessarily need an ILC to get decent video, manual controls or expandability. Sony's Cyber-shot DSC-HX30V and HX200V both deliver good video if you like your settings automatic. And the Panasonic Lumix DMC-FZ200 stands out on a few fronts: it not only supports manual exposure controls during video, but it also has a mic jack, a hot shoe (for a bank of LED lights), and its fixed f2.8 aperture across the entire zoom range gives you a lot more flexibility in low light than the typical snapshot camera.
There are quite a few cameras that I haven't yet tested that have potential to be good video options, including the Sony NEX-5R and NEX-6, Canon EOS 6D and EOS M, Sony Alpha SLT-A99, and the Panasonic Lumix DMC-GH3.
Sony Alpha NEX-F3
If you're looking for the cheapest ILC suited for shooting video, my choice is the F3, which runs about $600 for the typical body-plus-18-55mm lens. It's a nice little consumer-oriented camera with manual controls and very nice video quality, plus a lens lineup and autofocus system that works accurately, quickly, and quietly. While the video does display some artifacts, overall it's still better than a similarly priced camcorder; the biggest drawback is that it's interlaced at 1080/60i and there's no 1080/30p. You have to drop down to 24p for the highest-quality progressive option. Read the full review.
Canon EOS Rebel T3i
Last year's T3i still ranks as the cheapest dSLR for shooting video without autofocus. At about $600 for the basic kit, the T3i offers the high video quality and all the features an indie videographer on a budget needs, short of time code support and clean HDMI output. However, it has the typical dSLR problem with usable contrast autofocus, making it intrusive and frustrating to use. The (even older) T2i remains a fine choice as well if you can find it for less than the T3i; the T2i has reached the inevitable point in its life cycle where it's priced the same or even more than its successor. Read the full review.
Panasonic Lumix DMC-GH2
For the video geek on a budget who wants autofocus, this older top-of-the-line model is a great way to buy into a system that will scale with your interest. At about $700 with the basic 14-42mm kit lens, it's one of the cheaper options but delivers excellent video, full manual controls, and a mic input. The Micro Four Thirds mount has a nice selection of native lenses (including Panasonic's power zoom options and some great fast primes from Olympus), and with an adapter you can pretty much use any lens you'd like. To get the most out of the camera, though, you'll probably want to look into some of the firmware hacks available that deliver more features, increased bit rates, and better video quality. With the subsequent model, the GH3, Panasonic added in a lot of the capabilities provided by the hacks, but it also raised the price to about $1,300 for the body only. (I haven't had a chance to test it yet.) Read the full review.
Canon EOS Rebel T4i
As the least-expensive dSLR with decent video autofocus, the T4i makes a great choice for mainstream photographers who want a dSLR, but also want the ability to shoot typical video of kids, pets and vacations. The biggest drawback for this crowd is that "least-expensive" doesn't necessarily mean cheap. In order for the autofocus to work quickly and silently you have to use one of Canon's new STM line of lenses, of which there are only two at the moment, and which bump the price up significantly: the recommended kit with the 18-135mm STM lens will run you about $1,100. It's also a good choice for more advanced videographers who want the flexibility of access to a wide selection of Canon and EF-S compatible lenses for manual focus, with the occasional autofocus ability. Plus the touch-screen operation provides a nice interface for video recording. A more expensive option is the Sony Alpha SLT-A77V -- more expensive because it's a much better camera -- which produces good video and in auto has good autofocus. However, the A77V has too many video-related caveats, especially given its $1,700 price tag (with the excellent 16-50mm lens). Read the full review.
Canon EOS 5D Mark III
The overall best dSLR for video is currently the 5D Mark III, widely priced at about $3,300. It delivers the best tonal range, fewest artifacts, and, for the most part, broadest video-oriented feature set of the crowd. However, if you don't need hardware features like a headphone jack and you feel adventurous, you can get the also excellent -- and at around $1,800, a lot cheaper -- 5D Mark II. A bunch of the video features added in the Mark III, such as time code support and better audio control can be had in the Mark II by installing Magic Lantern, a third-party firmware hack (there's also an alpha version for the 5D Mark III that adds focus peaking and a few other features). The biggest hole in both camera's feature sets is clean uncompressed (4:2:2) HDMI out for recording to an external drive. Canon has announced a firmware upgrade for April 2013 that will supposedly supply this capability, but that's a long way off. Read the full review.