Watertight mesh: A watertight mesh is a 3D surface with no holes or gaps, so it fully encloses a volume and software can tell what is inside the part and what is outside.
Imagine filling the part with water. If it leaks anywhere, the mesh is not watertight. Every edge of every triangle has to meet another triangle exactly, leaving no openings.
This matters because a 3D printer, and any software that prices a print, needs to know what is solid. Volume, weight, material cost and print time all come from the enclosed space. With a hole in the surface there is no enclosed space, so the volume is undefined and the slicer has to guess where the walls are.
Example: an STL exported from a modelling program has a missing triangle where two surfaces failed to join, leaving a gap 0.2 mm wide. On screen the part looks perfect. The slicer, though, cannot fill the layers that pass through the gap and the print has missing walls, or the quoting tool reports no volume at all.
Watertight is often used loosely to mean “printable”. Strictly, a mesh can be watertight but still have problems: faces pointing inwards, or surfaces that pass through each other. The stricter condition is that the mesh is manifold and correctly oriented.
Most repair tools can close small holes automatically, but a repair is a guess at what the designer meant. When the fix is large, it is better to ask for a fresh export from CAD. Read design for 3D printing for other file checks worth doing.
Go deeper: Design for 3D printing: walls, overhangs, holes, clearances and orientation