Kevin's definitions and solution are good. Another way to look at this: what are the numbers that would make the shortest list of numbers, and satisfy this situation (mean & median=10, mode=8)?
If we want to ensure the mode = 8, then we must use 8 the most often: let's use it twice, so we start with 8, 8. The median (middle number) needs to be 10, so now we have 8, 8, 10. To ensure that 10 ends up in the middle, we need two more numbers, for a total of 5 (8, 8, 10, x, y). The mean needs to be 10, so what two numbers averaged with 8, 8 and 10 would give us an average of 10? we know the mean, or average is adding all 5 together and dividing them by 5, so we have (8+8+10+x+y) / 5 = 10. If we multiply both sides by 5, then we have 8 + 8 + 10 + x + y = 50.
There are probably many possible whole number (integer) combinations that would satisfy these conditions, but let's see if we can use numbers close to the ones we already have. Since we already have two 8's, We need numbers higher than 10 to give us an overall average (mean) of 10. Don't want to use 10 again, because we want to keep 8 as our mode.
We could try 11: 8+8+10+11+x = 50; 37 + x = 50, or x = 13
Now we have 8, 8, 10, 11, 13. Let's check these against what we need:
Most-used number = 8 = mode, check
Middle number = 10 = median, check
Average = 8+8+10+11+13 = 50, divided by 5 = 10 = mean, check