ASUS isn't well known for computer systems (at least in the USA), but they are very well known and well regarded for designing and building motherboards. A motherboard is the main board in a computer. Virtually every other component in a computer plugs into the motherboard. The motherboard is a complicated and essential component of any computer, and it kind of regulates every other component in your computer.

I have owned a computer with an ASUS motherboard and an ASUS graphics card, and had no problems with them, so I would not have any qualms about buying an ASUS computer. (All the other components in a computer will be made by other companies, anyway.)

The issue with the Canon Vixia HF 10 is that it shoots AVCHD video, which requires considerable "horsepower" to process smoothly. The bottleneck in processing AVCHD video is the CPU (central processing unit), otherwise thought of as the "brains" of a computer. In other words, to have a computer process AVCHD well, you need a sufficiently powerful CPU. (You also need enough memory but that aspect is trivial, and the computer you're considering has plenty.) The computer you asked about has the Intel E5400 CPU (Pentium dual core) which has a 2.7GHz clock frequency (the clock frequency is one of several aspects of significance). You can read a little about that CPU here.

My current computer has a CPU that is a little bit faster (2.8Ghz) and more powerful than the one in the computer you are considering. It is this one (Intel E7400 Core 2 Duo dual core). My computer is a little underpowered for editing AVCHD smoothly, but not so much that it bothers me. Different people have different tolerances for what they are willing to accept, though. If you are going to be editing AVCHD all day, day in and day out, my computer would be inadequate for the task, but I don't spend that much time editing. I do not have personal experience with the CPU that the computer you are considering has, but it is a given that it will be somewhat more underpowered for editing AVCHD than mine. My guess is that it might just barely be acceptable if you are pretty tolerant of jerkiness during editing (previewing). So, if you want smooth previewing, you will definitely need a faster/more powerful CPU.

Note: it's tricky trying to compare CPUs by their clock rates. They really need to be in the same CPU family to be comparable in that regard. But as bad as it is, that's the metric used for classifying their speeds. You can google tech sites that review various CPUs to find comparisons between different families and clock rates of them.