In English, the convention is to write the father's last name last: Obama is the last name.
The first name is the name chosen for the child at birth, the 'given name'. In this case it is 'Barack.'
We call the name between those two names the 'middle name'. Here it is Hussein. Often the middle name is omitted when referring to the person. People usually say 'Barack Obama' rather than 'Barack Hussein Obama'.
(In some cases the first name has two parts, such as Lucy Ann. You can't tell this from the way it is written. You learn this by noticing how people refer to that person, whether they address her as 'Lucy' or as 'Lucy Ann', for instance.)
If you're lucky, people insert a hyphen: -. This helps to distinguish between Double-Firstname Surname and Firstname Double-Surname. For example: Jean-Paul Sartre and Alain Robbe-Grillet. Those are two French people; British and American people do this less often .
The first name is also known as the given name, forename or Christian name. (Sometimes, the middle name is thought of as the second given name. The notion of the middle name is very strongly established in American English, but less so in other varieties.) What you call the second name is often called the last name, surname or family name.
Cagey has already given the first clue: the order in English (which is of course different from the order in Chinese). The other clue is that many English speakers have recognisable given names and surnames, so that if I see 'Margaret', I recognise it as a given name, and if I see 'Jones', I recognise it as a surname. In other words, some of it is cultural knowledge. The double-barrelled surname (whether hyphenated or not) are generally unusual, and you will be forgiven for not recognising it.