I'm new to Blender and just modeled my first object, a vial. Now after applying Subdivision Surface I've got this ugly seams and I have no idea why. Is it a problem of my mesh or what can I do about it?
I downloaded the file and found the issue. You had extruded faces but not moved them after that, and then having tiny redundant geometry and holes that messed with the subdivision surface modifier.
What I did to fix the problem was to select everything, press W and select "Remove Doubles", which should fix your issue.
You have vertices overlaying, select your whole mesh and press W > Remove Doubles and it will fix your topology. Play with the Merge Distance factor on the bottom of the T panel, on the left of the 3D view to make sure it will merge.