This would be Canon's AI Servo auto-focus, which is their continuous servo focus option. It is available on all Canon DSLR's as far as I am aware. The other two options are Single Shot and AI Focus. On higher-end Canon bodies, you have the option of configuring one of the back body buttons to focus. On newer Canon bodies, there is usually a dedicated AF-ON button on the back of the camera already configured for this purpose. Pressing and holding the AF-ON button (or whatever button you choose) when using AI Servo drive mode will indeed allow continuous shooting with inter-frame focusing, subject tracking, etc.
When it comes to subject tracking, the camera can do some of the work, but you have to make sure you keep the subject in-frame. You also have to make sure that the subject is within an appropriate range of the active AF points, otherwise the camera will be unable to properly determine what needs to be focused nor perform subject tracking. Proper continuous/servo AF technique with subject tracking and inter-frame focus adjustment requires skill, and just like any other skill, it has to be learned. You'll need to put in the practice to be able to effectively use any AF system, and the more hours, the better you'll get.
As for the highest AF points, it is actually Canon that has 61 points, on their two newest pro-grade cameras: The 1D X and 5D III. The Canon 61pt AF system is, spec wise, the best in the world at the moment, with 41 cross-type sensors and 5 double-cross type sensors, sensitive up to f/5.6. The Nikon AF system is 51pt, with 15 cross type, the center cluster of which is sensitive up to f/8, while the rest are f/5.6. Regarding whether one brand "beats" the other, that is truly irrelevant in the grand scheme of things. Both brands regularly leapfrog each other. At the moment Canon is the ISO and AF king (many early users of the 5D III have been amazed by the new 61pt AF system's capabilities and accuracy), where as Nikon is currently the Dynamic Range and Megapixel king. It is unlikely that these factors will remain the same for long, and after the next round of camera releases from both companies, the statistics will undoubtedly change again.
Don't use "who's best" as a factor for buying...that will never end well. Figure out what you want to do, what your budget is, whether you want to be able to share lenses with friends who have the same brand as you do, etc. The two Nikon cameras your friend is using, the D3s and D700, are very high-end cameras, and they cost a lot of money. Canon has similar cameras, like the 1D X, 5D III, 1D IV, etc. that also cost a fortune. If you are just starting in photography, you are looking WAY too high up the totem pole, and you couldn't possibly need a camera like any one of those listed right now. Look at the bottom range if you’re just starting. For Canon, that would be the Rebel series (xxxD numbers), and for Nikon, look at the D3xxx series. Buy a CHEAP camera body, as bodies come and go, and change every couple years. The true long-term value in photography is in lenses, and in that arena, Canon has a bit of an edge with a greater selection and a few very unique entries in their line-up, as well as some unique designs that no other company uses. Regardless, Nikon makes some excellent glass as well, and you can't go wrong with either brand.