'Round half up' is the most common rounding rule, and is the one that we learn in elementary school. Most people will never learn another way of rounding, but there are others.
Round half up is simply the rule that you identify the accuracy you wish to round to (e.g., nearest tens, nearest ones, nearest tenth), and then for the trailing digit (one place to the right), round values of 0-4 down to 0, and values of 5-9 up to ten. Thus, 3.4, rounded to the nearest whole number, becomes 3, while 3.5 becomes 4.
Because 0.5 is exactly between 1 and 0, rounding half up may introduce a bias towards increasing numerical values, particularly if you round frequently. This effect is small (assuming your numbers are all positive or all negative, you will have an average push away from zero of ~1%), but if you want to eliminate this bias, the most common solution is to round half to even.
It is relevant to note that while the common phrase 'round up' usually refers to round half up, in technical contexts it refers to rounding any trailing digit up, and is even more biased than round half up.