Ok, first thing's first. Let's get your terminology straight.
An atom's Valence refers to the number of bonds the atom typically forms in a molecule. I.e. hydrogen atom has a valence of 1 because it can form one bond with another atom (i.e. hydrogen fluoride H-F).
An atom's oxidation state refers to how the atom might behave in a particular molecule given its electron configuration. Oxygen, for example, almost always (except in peroxides...a story for another day) has an oxidation state of -2. In other words it hogs two electrons from other atoms
The column you referred to is called a group. A row is called a period (hence "periodic" table).
Unfortunately for you, you really do need to grasp the basic concept of electron shells in order to understand why certain atoms have multiple oxidation states and I strongly suggest reading these sections from your textbook before you go any further. But you also need to understand the concept of electronegativity. Electronegativity refers to the tendency of an atom to hog not just its own electrons, but the electrons of other atoms as well. The most electronegative atoms are located in the top right of the periodic table (i.e. O, F, etc.). Electronegativity tends to decrease as you move down and to the left of these elements. Oxygen, for example, almost always has an oxidation state of -2 because it is one of the most electronegative elements and therefore will always try to hog 2 electrons from other atoms in order to complete its octet.
Sulfur, on the other hand, is not as electronegative as oxygen (because it's lower down on the periodic table), even though both are located in the same group in the periodic table. We know that sulfur can have an oxidation state of -2 just like oxygen because it "wants" 2 more electrons in order to complete its outer electron shell. However consider the molecule sulfur dioxide (SO2): which element in this molecule is MORE electronegative? Oxygen. So which atoms are going to be hogging the electrons? The oxygens. Each oxygen atom will hog 2 electrons from the sulfur atom in order to complete its octet, and since there are 2 oxygen atoms in total in the molecule, the total number of electrons being "hogged" is 4 (2 oxygens x 2 electrons being hogged = 4). We know that elemental (neutral) sulfur has 6 valence electrons in its outer shell, but it will "lose" 4 of these to the oxygen atoms, giving it an oxidation state of +4.
In theory, nearly all elements towards the top right-hand side of the periodic table have multiple oxidation states, but the atom's electronegativity and other factors not discussed here have an influence on this. Chlorine (Cl), for example, has oxidation states of -1 (as you would expect), but also +1, +2, +3, +4, +5, +6 and +7.
My guess is that your teacher wants you to be able to recognize the most commonly observed oxidation states for certain elements. I highly doubt that he/she wants you to memorize EACH and EVERY single oxidation state which has ever been observed during chemical experiments for every element in the periodic table. What's important is that you understand how octets are formed and how atoms typically behave in bonds. And, not that it is any of my business, but in the future I would avoid trying to figure this stuff out at 4am the day before an exam. Just a thought...