You asked the right questions, but in the wrong order :p
What are you planning to do with the names they enter?
Searching (partial and full names), listing in alphabetical order (with or without grouping by first letter), and providing names for contact by telephone and other media.
What's wrong with just a "name" column?
Let's consider searching: if I want to search by surnames, a search of say, "Lee" would likely bring up people who have "Lee" as their given name. Likewise, a search of the given name "George" might bring up people who are surnamed "George". Separating the name into given name and surname is thus pretty much an instance of the principle of separating multi-valued fields into individual fields.
Likewise, if I were to group names by first letter, it is typical to want to group by the first letter of the surname as opposed to the first letter of the given name. Without separate fields for given name and surname, this becomes difficult.
I have forgotten where I first read about this, but more recent online sources for this principle include: Kexi's Introduction to Databases and ancestry.com's The Curse of the Common Name: Online Search Tips. For a published book source, I have "Beginning Database Design" by Clare Churcher (Apress, page 121).
Unfortunately, with cross cultural naming conventions the concept of a surname may not even be present. Even with surnames, generating a printed report can be tricky since we want ({name, surname}) {"George", Bush"} to be "Mr George Bush", but {"Ah Meng", "Tan"} to be "Mr Tan Ah Meng".
I am starting to suspect that the solution may be to just use it as a single multi-valued field, and then rely on a comma to separate name from surname in Western style names. It would take a little more effort, but at least then we can reasonably render "Bush, George" as "Mr George Bush" and "Tan Ah Meng" as "Mr Tan Ah Meng" while still being able to list by surname (or by first name, for those without surnames). The searching by surname problem would not be solved unless we are willing to allow duplication of values by keeping a surname field along with a full name field.
What do you think?