How the current method works

As Said before the current and most popular method of lawful execution is the Lethal Injection, this method is used in several states in the United States, of course other quick methods like hanging and lethal gas are used, the following picture shows what States use the following, keep in mind this is only for the United States of America.

The United States became the first country to use the lethal injection in the year 1982 Oklahoma became the first state to adopt a lethal injection legislation. of the 38 States that use Lethal injection, approximately 34 use it as a primary mean of execution, leaving the other methods behind.
As of 1994 this method became more adept to using the lethal injection, it increased from 27 to 37 on 2004.

Other countries began using the lethal injection method in 1997, China become a popular one for its many executions. Other countries like the Philippines, Guatemala and Taiwan also adopted This method around the same year. The current process used in China is most likely not similar, They keep their methods secret and it is hard to obtain any information at this current time.
To date Approximately 120 countries have a sort of Capital punishment, but 20 of these have not executed a convict in the last 10 years or more.