Tube diameter is that diameter on the rifle scope, where mounts are clamped on (see the picture below).
This parameter is an important information when buying proper mounting rings for your rifle scope. Most common mounts have a diameter of 1 inch, 30 mm and 34 mm, but in the last years, manufacturers started to produce rifle scopes with a tube diameter of 35 mm, 36 mm and even 40 mm to increase windage and elevation travel range.
Lower price ranking and older rifle scopes have a 1 inch (25.4 mm) main tube.
Common for hunting rifle scopes is a tube diameter of 30 mm, which is very tough for hunting use and for rifle scopes with a BDC turret the scope has enough elevation and windage travel range.
Most tactical and some sport rifle scopes have the tube diameter of 34 mm, because of a thicker wall (of the tube) for a more reliable use in a harsh environment or an extended elevation range against 30 mm main tube rifle scopes.
Some of the new rifle scopes (for example Zeiss victory V8) have an unusual 36 mm tube diameter, also some IOR or the new Swarovski DS rifle scope stands out with a 40 mm tube diameter. European hunting rifle scopes have often rails, where mounts clamp on these rails and not around the tube. This ensures a tension free mounting, without any marks on the rifle scope itself.