Simplifying a fraction makes the numbers smaller while keeping the same value, and finding a common denominator makes fractions with different denominators share the same one.
8/12 and 2/3 look completely different, but they're actually equal in value. Dividing the numerator and denominator by the same number makes the numbers smaller without changing the value the fraction represents at all — this is called simplifying. Dividing both 8 and 12 by 4 gives 2/3, and dividing by their greatest common divisor all at once gets you straight to the simplest form that can't be divided any further.
Fractions with different denominators are hard to compare directly. So we multiply the numerator and denominator by the same number to make the denominators match — this process is called finding a common denominator. Watch the bar diagrams and see for yourself that the shaded proportion stays exactly the same before and after simplifying or finding a common denominator.