Each NPC reacts differently to each selection.

You have (in my phrasing):

Love it (BIG Smile, Eyes wide)
Enjoyable (slight smile)
Disdain (slight frown)
Hate it (deep frown, angry look, narrow eyes, furrowed brow)


Object is to get them to like you better (usually, though you can get them to fight you on occasion). Do this by selecting the big slices on the first two and the small slices on the second two. If you want them to hate you, do the opposite.

After skill level 25 (or is it 50?) you will have the option to make the wheel skip ahead once without a selection, but that is the only time you can do it.

For a lot of quests, you have to gather information before you can do what needs doing. Most people wont tell you and sometimes there isn't even a selection until you raise their liking to 80 or 90 (out of 100). There is usually another way to get the info, but this is also usually the easiest. Also, you can get better prices when you haggle with the higher likings, for both buying and selling goods. And you definitely want that!

EDIT: It can be hard to tell the differences in expressions on the Khajit and Argonians, it just takes practice to recognize with them.

EDIT 2: Practice on guards, as they really won't do anything differently and you never know when a normal NPC is going to be useful in a quest.

EDIT 3: some grammar but also, it might be obvious at this point, but the bigger the slice, the greater the reaction, either positive or negative.