The formal name is 'snap cap'. And it's a PITA.
Like Kevin said, its the retainer that fits inside the cap. The female recess on the small end (at the top in your picture) screws onto a threaded stud under the cap finial. Then, the slightly proud rim around the end of the nib section of the pen fits into the other end and 'snaps' into place, holding the cap on the pen body.
First problem is that the recess that is supposed to screw onto the threaded stud isn't threaded. So you have a choice of either forcing it to screw onto the stud, or using an appropriate tap (it's almost certainly metric) to thread the recess. In my experience, the former approach is difficult to do (and its easy to damage the retainer when doing it), and the latter leaves theads that are so loose that the retainer can easily unscrew from the threaded stud. In theory, a drop of CA or Locktite in the threaded recess should solve that problem, but that fix can be messy.
Second problem is that the retainer fits deep inside the cap, and you need a way to grip it while you screw onto the threaded stud. In general, the answer is a screwdriver. My experience is that a No 2 Phillips screwdriver is better than a flat blade screwdriver, and even better is a No 2 Torx screwdriver. But the best device for inserting these is a No 2 Phillips screwdriver with a mm or so of the tip ground off leaving a cross-shaped profile with four sharp points to grip the inside of the retainer as it is being screwed onto that threaded stud.
Third, the retainer has to be screwed far enough onto the threaded stud for the pen nib to be captured in the retainer. But if you screw the retainer too far onto the stud, the retainer will be too deep inside the cap, and as a result, the section can fit far enough into the retainer to snap into place. Conversely, if you don't screw it far enough onto the stud, the section will snap into the retainer, but there will be an unsightly gap between the cap and pen body. Getting it just right is tricky.
Finally, this is a fountain pen, and in addition to being the retainer that holds the cap on the pen body, it is also supposed to seal the nib and feed to prevent them from drying out when the pen is not being used. Unfortunately, snap caps don't seal very well and as a result, pens that use this kind of closing mechanism tend to dry out more quickly than pens with a screw cap.
Hint: most kit instructions tell you to install the cap finial, and then use a screwdriver to insert the retainer into the cap and screw it onto the threaded stud. My experience is that it is better to fit the retainer to the stud before installing the cap finial. Then, when the cap finial has been installed, use a screwdriver to adjust the position of the retainer inside the cap so that the section snaps properly into place. If the retainer is only loosely screwed onto the stud, you may have to remove it entirely, apply CA or Locktite, and then reinstall it, being careful to not get the CA or Locktite on the inside of the cap tube.
I'm a devoted fountain pen user, and while I have made a few pens with these 'snap caps', I much prefer pen kits that are designed for the cap to screw onto the body.
i had to google PITA and scroll down all the pita bread images..