In the following video, there is an explanation of asymptotic analysis: https://class.coursera.org/algo-004/lecture/169
But I can't understand what is "low-order term" and "constant factor" itself? (it is at the 4th minute of the video).
The merge sort is 6n*log2(n)+6n. Why 6n is the low-order term and 6 is constant factor in here? Do these terms have concrete definition?
Lower-order term:
"Order" here refers to the order of magnitude.
The concept is easy to understand and explain when dealing with very simple terms such as x or x2. x has order of magnitude 1, since it can be written as x1, and x2 has order 2 - the order of magnitude is equal to the power of the variable in the term. But things get a little more hazy (at least for me) when you complicate things by, for example, adding log. [1]
In somewhat informal terms, f(x) is a lower order than g(x) if f(x) < g(x) as x tends to infinity.
It's easy to see that f(n) = 6n is a lower order than g(n) = 6n*log2(n) by just substituting some really large value for n (the correct approach is to mathematically prove it, but substituting a large value tends to work for simple terms).
The terms are essentially the things separated by plus / minus symbols.
So a lower-order term is just any term which is of a lower order than some other term.
Presumably this is the opposite of the leading-order term, which is the term with the largest order of magnitude.
[1]: I deal with big-O a lot, but it's been a while (high-school?) since I've dealt with the basics of order of magnitude, so apologies if I might have missed or forgotten something regarding that part.
Constant factor:
"Factor" is a term in multiplication. For 6n, 6 and n are factors.
A constant factor is simply anything that doesn't depend on the input parameter(s) (which is n in this case).
Here, regardless of what we make n, 6 will always stay 6, so it's constant.