Armband tattoos are a common type of tattoo, especially in tribal or Celtic styles. It's a single unbroken strip of tattooing going all the way around the arm. It's also possible to do legband or wristband tattoos.
Some tattoo artists will tell you that "it's bad luck" to fully connect the design, but that's often just to cover up for ineptitude, since it takes some skill (and work) to scale the design to perfectly fit an arm if the design is to be a single repeating pattern that totally encircles the arm. Some artists may also use a distinct centerpiece to act as the connection point for the tattoo.