so could I get any digital 3d model and just print it?
No. There’s always at least the minimum wall thickness of the printer to take into consideration. Most 3d print methods have more limitations such as limited overhang/bridging or inability to print one loose object inside another (like the ball in a whistle) that prevents some geometries from printing.
You also need to ensure your model is watertight…you’ll also see terms like manifold used. The basic idea is that if you filled the internal spaces of the model with water, would it leak out? If your model is non-manifold, it will leak and not be printable. There ARE tools to address this (MeshLab, etc.), but you still need to keep in mind some of the items Hannes mentioned…particularly overhangs and bridging. These two problems can often be fixed by letting your (slicing) software add automated support structures to the model that are then snapped off post-print.
More info on watertight models: http://airwolf3d.com/wiki/manifolded-edges-how-to-check-if-your-model-is-watertight-and-ready-to-3d-print/