When a cellular phone makes the designation that it's in "Roaming Mode" it means that the phone is unable to connect to the service provider's cellular network, but it can connect to another cellular provider's network. Roaming is usually associated with using the cellular phone outside of the service provider's network area, but it can also be tripped by having a communication error with the closest cellular towers, when local towers are down for service or when a network is running at capacity.
Roaming Advantages
The advantage to roaming is that your cellular device will be able to make calls and access the Internet in the event your phone can't access the service provider's network. If there are no cellular towers in range of a phone, it is unable to receive service. By enabling roaming, cellular service providers are able to extend the area that their customers can be in and still use their devices. Without roaming, a person who goes to a town that isn't covered by their service provider would be unable to use their phone to place a call or use mobile Internet.
Charges
You may be charged for using your phone while roaming, depending on your service plan. Some carriers are able to avoid charging customers who use their phones while roaming, because of service-sharing agreements with other carriers. Phone calls use relatively little bandwidth compared to data access, so carriers may consider the difference negligible and ignore charging customers for calling when roaming. Data, however, uses up a lot of bandwidth and carriers may not be willing to share it, which may lead to additional charges.
Call Roaming and Data Roaming
Roaming usually implies that the phone is entirely running off of another network, rather than the service provider's, but sometimes it may only be partially roaming. There are two subsections of roaming: call roaming and data roaming which pertain to making phone calls and using mobile Internet access respectively. Call and data roaming are important to differentiate because some roaming plans may only cover making calls and sending text messages as opposed to using the data network to access the Internet. Call roaming and data roaming may also be tripped individually, if a local cellular tower on your device's native network is at capacity in either calling or data.
Disable Roaming
Most cellular phones offer some way to disable roaming data usage to avoid additional charges. Disabling roaming can be useful when traveling internationally, because international roaming charges can be substantial, compared to domestic ones. Additionally, you can be charged for roaming even if you turn off the device, because even when powered off, the phone may run in a low power mode and still maintain a connection with the nearby cellular towers.