The Data roaming setting enables or disables the use of mobile data while roaming.
Roaming is using your phone via a cellular connection through a provider other than your own, due to a roaming agreement between the two providers.
Roaming agreements between carriers allow customers to travel outside the coverage area of their own carrier and still receive a mobile connection, allowing them to make and receive calls and send and receive messages. Most carriers will have roaming agreements with selected carriers around the world, allowing for their customers to use their phones while travelling internationally.
However, such usage is often at highly inflated prices. This is especially true for Internet data usage, where usage while roaming internationally may cost 10, 100 or even 1,000 times as much as it does in the customer's home country.
For this reason, it can be dangerous to use a smartphone or other internet-enabled device when roaming internationally even while trying to conserve your data usage, and Android will usually disable your data connection while you are roaming by default.
Note that this setting does not affect your ability to roam; only whether data is enabled or disabled while roaming. You will still be able to make and receive calls, and send and receive text messages, while this is turned off, and this is still likely to cost more than it does in your home country.
Occasionally, a carrier may provide special roaming deals whereby customers can use data while roaming cheaply. If you have such a deal, you may consider enabling this setting. You should check carefully its conditions first.