Mobile phones were handheld mobile communication devices. On the planet Earth, they became prominent amongst much of the population in the early 21st century. With many of these devices, people could view the person they were talking to on screen.
Rose Tyler, Martha Jones and Donna Noble had their mobiles adjusted by the Doctor (Ninth and Tenth Doctors respectively) to have 'Universal Roaming'. These became known as Superphones as they were able to call anywhere in time and space and could apparently work in areas of no service, such as the Cabinet Room. The Doctor also modified Barclay's phone in this manner, but he did it so he could communicate with UNIT. It is unknown what he did with this phone afterwards, if he undid the Universal Roming and gave it back, just gave it back as is, or kept it. (DW: The End of the World, 42,The Doctor's Daughter)
Anji Kapoor carried her mobile with her when she began travelling with the Doctor, and was even able to use it when she travelled into her future. (EDA: Escape Velocity, Trading Futures)
The Eighth Doctor also set up mobile phone-like devices for Ani and Fitz Kreiner when they spent a selective amount of time on certain planets. (EDA: The Book of the Still)
The Ninth Doctor was shown to have his own cellphone in (DW: Boom Town), but it was never seen again. The Tenth Doctor would later keep another phone given to him by Martha Jones. (DW: Last of the Time Lords)
Sarah Jane Smith, Josh Townsend, Natalie and their respective friends frequently used mobile phones to keep in regular contact with one another. (BFA: Sarah Jane Smith (audio series))
By 2050, mobile phones appeared to have been replaced by the Vid-com. (K9TV: Liberation, The Cambridge Spy, Angel of the North)