What is Share Buyback?
It is the company buying back shares of itself to reduce the number of shares outstanding, and in return increase the value each share is worth.
For a more in-depth explanation about it, refer to the Investopedia page HERE.

Why Companies (particularly US companies) Love Share Buyback?
The main reason: Tax
In the US and other countries, shareholders have to pay a dividend tax for the dividends they receive.
In the US, the nominal tax rate is 30%.
This causes shareholders to experience double taxation - once on the company's profits, then again on the dividends paid out of the profits.
To avoid double taxation, companies use share buybacks as a way of "giving" returns back to shareholders.

The math of Share Buybacks
You can skip this if you know how buyback works
A company made $1 million in profits, has 1 million shares outstanding, and a PE of 10.
The current share price of the company is...