Clarifying the meaning of market capitalization I'm new to this kind of topic and I'm trying to get a deeper understand. 
I read that "the shares' price times the number of shares outstanding is the value of a company (=market capitalization)".

However, I'd say that it's just the value of that part of the company that is in the stock market. 
In other words, when a company issues an IPO, it might not issue ALL of its shares, but just some of them.
So, let's say that the total amount of shares of a company (A) is 1000. The outstanding shares are 300 at the price of 10\$. The market cap is 300x10=3000\$. This number is the cumulative value of all the issued stocks.

Let's take this a bit further by taking another company (B). The second company has a total amount of share of 10000 (ten times that of A). So, if B issues 100 shares at the price of 10\$ its market cap will be 1000\$.

By comparing the market cap of A and B I'd say that the value of A is three times greater than B, however, that's not correct. Or I should say that it is indeed correct for what regards the market cap, but that's not the case if we consider the shares that have not been issued yet.

So the question is, how can the market capitalization give a hint about the value of a company? Also, how can the market cap be used to compare companies without taking into account the ratio between outstanding shares and the total amount of shares? (like the example above)
The "shares outstanding" variable used in the calculation of market capitalisation is defined as all shares that exist (and not was issued in public offerings). The market capitalisation is the share price times the outstanding. Therefore, the numbers in question do not apply.

There is no need for shares to be transferred in a public offering; an initial investor in a firm before it was listed could sell to another investor privately. This means that there is no good way of tracking how many shares are held by "the public," and that number is not used in calculations.