Update: Some of the equipment described in this otherwise excellent answer has been updated. In 2014, Canon replaced the EOS 7D described below with the 7D Mark II, offering a number of improvements over the original version:

• 65 point AF all cross-type AF system (up from 19 AF points)
• improved AF tracking
• 10 frames/second in continuous drive (up from 8)
• larger memory buffer allows more continuous shots (31 in RAW mode, up from 25, and many more in JPEG)
• 20.2MP sensor that uses Canon's "dual pixel" CMOS technology
• improved sensitivity (max ISO 25600 up from 12800)
• built-in GPS

Also, in late 2014, Canon introduced the EF 100-400 F4.5-5.6L IS II USM which offers a number of improvements over the version described above, including a rotating zoom ring (vs. push to zoom), even better image quality, a third image stabilization mode, and (naturally) a higher price tag. As of early 2016, the original EF 100-400 f/4.5-5.6L IS USM described below is still in the lineup, however.

---

From personal experience, I recommend the following camera body: The Canon EOS 7D.

I say this for four main reasons:

1. It is an APS-C crop sensor camera, which in Canon terms gives you an automatic 1.6x multiplier, allowing you to get closer to your subjects.
2. 19-point advanced Auto Focus system.
3. Fast 8fps burst rate, even with RAW.
4. New (at the time it was introduced) 63-zone iFCL metering

So with the crop, you can stay where you are normally but your lenses will all have the 1.6 multiplication factor applied. For animal and bird photography - you want the extra reach the crop gives you over full frame.

Secondly, the AF system on the Canon 7D is ridiculously good. I took this photo was achieved with the spot AF mode, but it also has a 'zone' AF which is like full-auto except you restrict it to one part of the frame. There is also a single-point expansion mode which is like single point but allows the AF to be a little intelligent regarding what's in the range of the AF points immediately adjacent to the one selected so if your subject moves just an inch the AF still sees it and locks on. Then you have the standard sort of full-auto, single point modes etc. The other good thing about the Canon 7D's AF system is its tracking abilities. In AI Servo mode you select your starting AF point and it locks on. If the subject moves within the frame, the AF automatically tracks it. The 7D is HIGHLY flexible in the way its AF works and there are options to select say, the sensitivity of it to subjects that may get in the way. For example, if you are following a bird, and it flies behind a telephone pole, or power line, or so forth, you can tell the AF to be 'slow' in its responsiveness to this kind of thing so it doesn't go out of focus immediately. Likewise if you were shooting something else you may wish it to react quickly to such 'interruptions'. This is all totally configurable on the 7D.

Thirdly, the burst rate. You will never achieve perfect shots 100% of the time. It just wont happen. Blurries and missed shots are an inevitability of shooting fast moving subjects whether they be birds or other animals. With 8fps sustained burst rate, you can follow your subject with the tracking AF as described above, with your finger pressed firmly on the shutter button and capture many many images all very quickly. You can then decide later which ones to keep and which ones to throw away. With animals, too much choice is better than not enough! Note that the 7D also gives you configurability when it comes to the trade-off between autofocusing and shutter speed.

I will admit, the 7D has some detractors regarding the AF system, because they don't necessarily get the shots they expect. But it is a highly customisable system, and I really recommend reading and understanding the manual because once you understand it and set it the way you want it, it will reward your efforts.

Lastly, the metering. The 7D has a new metering system and this comes in very handy if you are shooting against the sky. You will want to use spot metering if you are following a bird across the sky. It performs really well.

With regards to the lens, it is difficult to make such a recommendation because it is entirely subjective. Some would advocate the EF 70-200 f/2.8L II IS USM, whereas some would advocate the f/4 version. Some would tender that if budget is no option then go for a 300/400/600/800mm f/4 or f/5.6 telephoto. Whilst these are all GREAT lenses, they are out of the reach financially of many. And to be honest, any good lenses are going to be expensive. You are not going to achieve great shots with a £170 75-300 f/4.5-5.6! However I can tell you what I use which I think is a good tradeoff between price, quality, and reach.

That lens is the EF 100-400mm f/4.5-5.6L IS USM. New, it retails for somewhere in the region of £1250, but I got mine on eBay second hand for around £820. A bargain really considering what you can pay for prime telephotos!

The 100-400 is in Canon's "L" series lineup, and really is a great quality lens, and I am very pleased with it. It's big, and gets even bigger when you zoom to its full reach, thanks to the slightly odd 'push-pull' zoom mechanism. Some people find this weird, but from my experience, you get used to it VERY quickly.

It's a slow-ish lens, so not great for low-light or night shooting unless tripod mounted (it takes GREAT moon photos!). But for daytime shooting, it has zero issues, and the IS really helps to maintain steadiness too which helps especially at the longer end of the zoom range. 100 to 400, which on the 7D is equivalent to 160mm to 640mm so gives you a nice range with which to work. It has a focus limit switch on it which restricts the MFD to 6.5 metres (I think?) which makes the AF much faster when shooting things at a distance as it wont even attempt to 'hunt' below this distance.

Anyway, that's enough talking. To demonstrate my points above, here are some example photos that I took, with the combo of the 7D, and 100-400. A couple of birds, and a couple of planes. (one could argue they are birds too!):

Gull

Another Gull

Duck landing on water

Hawker Hurricane

F18-A Super Hornet

I hope that helps you out...