I'll try to explain the whole thing while answering your questions.
Where do you put your "Standard Array" number. In score? Abil Mod? Mod + 1/2 Lvl?
In the "Standard Array" you have 6 numbers you can rearrange to fit your character's needings. These numbers, once rearranged, will be your scores.
For example, I can say «my PC's Strength score is 15».
What is MOD? (As my concern, Abil Mod is the table of chart from -5 to +5)
Mod stands for modifier and it's the number that you'll effectively use in your Ability Checks (added to the half of your level most of the times). It gives a quick representation of your PC's abilities. While you can use the chart to determine your modifiers, you can use this easy expression:
Modifier = (Score - 10) / 2 [Rounded down to the closest integer number]
For example, I can say «my PC's Strength score is 15, so his Strength modifier is 2».
Modifier = (15-10)/2 = 5/2 = 2.5 = 2 (rounded down)
In this case, while filling your sheet, you must write (assuming your PC is level 1) 15 in score, +2 in mod and +2 in mod + 1/2 lv (because 2 + 0.5 = 2, rounded down).
Note that 10 and 11 have 0 as a modifier. In fact, 10 is the average score for a common person.
In Fort, Ref, Will, do I put my total of score, abil mod, or mod + 1/2 lvl
You should fill all the fields, if you have a bonus.
Let's try with Fortitude, always assuming you are Level 1. In the great circle you will have to write the total, let's skip it for now.
In 10 + 1/2 lv you would write 10 (because 10 + 0.5 = 10); In Ability, you would write your Strength/Constitution modifier, whichever the best. Let's say you have 12 Constitution (+1 modifier): you will only consider the Strength one (since you have 15 Str), so we will write 2.
The rest (in order) stands for Class, Feat, Enhancement, Miscellaneous, and must be filled only if you have some bonuses in it. Assuming we don't have any of those, our final Fortitude score is 12 (10+2);
How many healing surges do I have? It says Ability Modifier, but in the score table, abil mod or mod + 1/2 lvl?
If it says Ability Modifier, then it's Mod :P
Finally. In skill, what do I write in the Abil Mod + 1/2 lvl?
Like above, you need to write the Ability Modifier plus half of your level (for example, you would write 2 in Athletics).
Follow-Up Questions Edit
I'm human. I can add +2 to one ability. Where should I put it, will it modify the score and the ability modifier?
You usually choose the ability depending on your class. For example, if you are a Fighter, you will need more Strength or Constitution. That +2 gets added to your score. Given the previous example, your final Strength score would be 17 (+3 modifier).
Remember that all the spells/feats/whatsoever affects your abilities, affects your score. From the new score, you can calculate your new modifier (+3 in this case).
Since I'm human, my healing surge is 6 + constitution modifier. My Cons abil mod is 1, will it means that I got 7?
Healing Surges are usually given by classes. Are you sure you didn't mean Warlock/Wizard/Rogue/Ranger? However, yeah, the result is correct! :)