What is the diffrence between those two words.

I know that 'elect' is used when you are talking about politicans and presidens like George W. Bush.

And select is used more common when you have a choice.

But is that the right defination of the meaning of these two words?
Yes, you have the correct definitions in mind. We use elect when a group of people vote. To elect is normally to select by means of voting. We use select to mean choose. Occasionally, in more formal or official situations, elect is used to mean select. For example, you may be asked by your employer to elect one or another insurance option (one of your employee benefits). In this case, no voting is done, and elect is just an official way of saying select or choose. Even more informally, pick also means select.

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On the website: http://www.espncricinfo.com
we see
England won the toss and elected to field.

Here the word elected is used because selection is made through a process (Toss).
The word select is used when we have right/authority to choose.

Am I right?
I agree. It appears to me that in this sense, "elect" is used as a more formal term for "select."

"Select" might mean to choose from a list of standard options, whereas "elect" requires you to be formally presented with a choice rather than that choice having existed previously. This is the only difference I can find between the two words in this sense -- it seems to be hinging on whether the candidates existed before the act of choosing was initiated.

If, before the toss was conducted, each team was required to choose to kick or field, then perhaps that would be more accurately described as a selection.

For instance, in the US, the term "election" is often used for health coverage, because it's a formal process and the decisions made are often based on the options that are presented, which may very from one individual to another -- you can't really pre-select your coverage because you may not be eligible for all of the options.

This is why language is such an interesting subject, and an example of how healthcare in my country is silly.