A GCF is the greatest common factor of two numbers. It's also sometimes called the GCD, or greatest common divisor. It's the largest number that two numbers can both be divided evenly by ("goes in to"). This number is useful when you are trying to simplify fractions because you would divide by a Giant One. For example, 36/48. The GCF of 36 and 48 is 12. Divide it by 12/12 and you get the simplified equivalent fraction of 3/4.
The LCM is the least common multiple. It's frequent use is to find the LCD or least common denominator of two unlike fractions. For example, if you wanted to add 1/12 to 1/15, you'd have to find the LCM of the denominators, 12 and 15. That's 60 so 1/12(5/5) and 1/15(4/4) gets you 5/60 + 4/60 which equals 9/60 or 3/20.
Here is a method a student taught me to find the LCM of larger numbers. This student clearly understood it: