The coefficient is the number written to the left of the variable. It is a multiplier, so 3X^2 means X^2 times X^2 times X^2. Coefficients are used in the same roles in chemistry where 3O2 means three O2 molecules. The coefficients are sometimes written with letters in statements intended to express ideas that are true for any coefficients -- and in this role the coefficients are often taken from the front of the alphabet -- a, b, c, while variables are usually taken from the back of the alphabet -- x, y or z. For example in the statement : If aX^2 + bX + c = 0" the understanding is that a, b and c can be any coefficients.
Subscripts are the small numbers that appear after and slightly below the variable. In mathematics, subscripts are usually used in an indexing role. For example, if A is an array, writing A with subscripts 1,3 would be a reference to the element in row 1 and column 3 of the array. In chemistry, the subscript refers to how many atoms go into the molecule. The normal component of gaseous oxygen is O with a subscript 2. Ozone is O with a subsccript of 3. So 5 ozone molecules would be an O with a coefficient of 5 and a subscript of 3.
Superscripts are like subscripts except the are slightly raised instead of lowered. In mathematics, superscripts indicate exponents. So 3 times X squared would be X written with a coefficient of 3 and a superscript of 2.
Coefficients, subscripts and superscripts all have their own conventions. When the coefficient is 1, it is not written. When used to indicate exponentiation, a superscript of 1 is not written. The polynomial aX^2 + bX + c is equivalent to aX^2 + bX^1 + cX^0, but it is never written that way. The conventions make the expressions more concise. As long as everybody understands the conventions there is no ambiguity.