If I interpret your question correctly, you are defining "the flower" as what was originally the
base of the flower, before the petals fell off. This is called the stem of the flower, and becomes the stem of the pod. There may be some confusion because a) the flower stem is not very long on most pepper plants, and b) it starts out very slender (though in some cases it becomes quite thick as the pod develops.)
Since you were using unfamiliar terminology, some of the answers above might be confusing for you, in turn. As 3/5King stated, clip the stem of the pod where it joins the rest of the plant. Alternately, as sicman stated, you can slowly turn the pod and see if it just pops off. If it's not ready, the stem will twist, but not easily come off the plant.
If you are going to cut the pods off, a hand-held pruner works fine -
here's a link to an example of a pruner. You can find less expensive ones than the ones I linked to, but that's just for a visual (though I like the slender tip on that pair!)