The Disposition Mini-Game is hard to understand at first because there is no proper explanation either in the manual or in-game but once you get it its a breeze. Here I'll try to explain it the best I can.
There are 4 "types" of disposition ; Admire, Joke, Coerce and Boast (in order or apparence clockwise in the mini-game circle starting from the top). There is 4 type of appreciation level ; Like alot, Like, Dislike, Dislike Alot. And finaly they are 4 size of "gauge" from full to little. The first thing to do when you start the mini-game (and by start I mean you need to have pressed the "Start" button) is to check which Types they appreciate the most and which Types they dislike the most, to know that you just need to hover your cursor over the types, the facial expression will change following their like/dislike, this is normally easy to see on human character but some non-human character can get a bit tricky but dont worry you'll get the hang of it.
What you want to to do is to left clic in the Disposition Type portion of a Type the character really like when the gauge is big and clic in the Disposition type portion when the gauge is small. The mini-game is divide in "rounds" where in each rounds you have to select all the four disposition. Each time you clic one type the gauge all move clockwise and the Type selected gray out, you wont be able to re-select this type until next round.
Okay lets try with a 'sorta' graphical exemple;
Here we have Mr.Bob , Mr.Bob Like alot jokes, Like Boast, Dislike Coerce and Dislike Alot to be Admire (go wonder). Lets try to make a round shall we. The order is the one you would get clockwise. 1= smallest gauge, 2= small gauge, 3= big gauge, 4= full gauge
The logical choice here would be to select "Boast" as "Joke" have a very small gauge and the two "dislike" have bigger gauge then we want. Lets choose Boast.
Boast: 2 ----Gray
Now that we selected one of them the gauge all turned clockwise, as we can see the biggest gauge is on Joke, which he love, now of course we should select this ---- beep, wrong. If we select Joke now that mean that during the next round we'll have to choose between either 2 on Admire or 4 on Coerce and then final round between 1 on Admire and 3 on Coerce, whatever the order you would choose, this rounds would make you loose disposition instead of winning some because you Loose more then you Win with the biggest gauge when you dont have a lot of Speechcraft! If you were a Journeyman you could have done that but if you're not dont. The best choice would be to click Coerce now because it doesnt decrease by alot the disposition when you have the smallest bar.
Coerce: 4 ---Gray
Boast: 1 ----Gray
Now as you can see you still get a nice bonus if you select Joke, its better to select a gauge a bit smaller then the full gauge on a "Like" then selecting a full gauge and having to choose for a bigger then smallest one on a "dislike".
And there you select the final one "Admire", congratulation you just increased the disposition of the NPC by a bit and increased your experience in speechcraft.
Once you get "Journeyman" skill level in Speechcraft you'll have the option to "rotate" the gauge clockwise once in each round. I hope you got it