The Independent Chip Model (ICM) is a basic concept, to be a longterm winner in sit and go (SNG) and Multi Table (MTT) Poker Tournaments.
Players, who do not know this concept, always make decisions, which are not profitable, or not that much profitable as supposed.
You can see horrible plays from these players, especially on the bubble.
The Independent Chip Model is representing your current equity in a prize pool based on the relative stack sizes of the remaining players and the distribution of that prize pool among them.

How we can use this?
1.You play a big MTT and your opponents offer you a chipcountdeal.
Always good to know if this deal is fair, right? So why not calculate your equity with the ICM and see if this deal is good for you.

2.You have to decide to go all in or to call an all in by another player on bubble.
ICM is the best information in these situations.

The mathematical basics

In the model you assume from the fact that the chance to win a tournament is as high, as the number of your own chips relative to all chips available during the tournament.
Example:
Four players remain in a Sit & Go with which ten players have begun with 1'000 chips in and bought in with 10$ each.

If you know what your current equity is you can use this to make decisions based on how much extra equity you will either gain or lose by taking certain actions (usually pushing all in or calling an all in at this stage).
For example if you have 24% equity to win and you will lose all of this by calling an all-in,
you need to balance this with what equity you potentially gain by winning the hand.
This is often less than you stand to lose – for example doubling up might increase your equity by 24%.
Now instead of counting your winning chances against the range of hands your opponent could have pushed with,
you instead count your equity risk against equity reward then look the range of your opponents hands compared to your holding.
If you think you are 60% favorite against your opponents range but are risking $24 to win $12 then it makes sense to fold your hand – the equity risked is not compensated by the potential gain in this situation.
This is the key to use the Independent Chip Model:

If you take positive expectation situations you will win over time.
Conversely if you take negative expectation situations you will lose...

Lets put it in numbers to be even clearer!

You have 24% equity and are 60/40 favorite (an estimate of course) to your opponents range when he pushes all in ahead of you.
If you win and double up you gain $12 additional equity, if you lose you are out in 4th (equity = $0).
So after 100 hands....
60 times you win and have $36 equity 60*36 = $2160
40 times you lose and have $0 equity.40*0 = $0

So at the end of 100 tries you have $2160/100 or $21.6 in equity - the play costs you $2.40c each time you make it when compared to folding!

How is Prize Pool Equity Calculated?

Each remaining player at the bubble has a chance to finish in any of the 4 remaining positions...
To work out current $ equity we need to calculate the chance of each position for the 4 players.
The 4 players have stacks of:

Player A: 5'000 chips
player B: 2'500 chips
player C: 1'500 chips
player D: 1'000 chips

Here is how it is done
We start with the big stack, based on chips alone he has a 50% chance of finishing 1st and taking the $50 for 1st place, that is $25 in equity.
Next we assume that the big stack indeed got 1st and now we want to know which chance the other players have in finishing 2nd / 3rd / 4th (based on % of chips in play after we exclude the big stacks chips).
So Player B now has 50% of the remaining chips and thus a 50% chance of finishing 2nd.
If we go on with this Player C has a chance of 60% to get 3rd and so on.
Next we start to look at Player B´s chance in finishing 1st (25%) and then work out the other stacks chances of taking 2nd / 3rd / 4th based on their chip stacks.

At the end you have worked out each stack’s chances in finishing in each position and now you are able to calculate their total equity as an average of their chance of each finishing position and the payout for that position.

If you take one thing from this section it should be that doubling your chip stack does not usually double your prize pool equity...
this fact is most important to do the next step - using ICM at the poker table!