Get a hammer drill with a carbide drill bit. If you do not plan on drilling holes into concrete very often, rent a hammer drill from your local hardware store.
Place the handrail where you'd like it be and mark through the screw holes with chalk. Remove the handrail and then, using steady pressure -- but not pushing with all your might -- drill holes into the mark with hammer drill. To achieve a certain depth, either use the depth gauge if your drill has one or mark the correct depth right onto the bit.
Remove any dust from within the holes by blowing compressed air into the holes. Any remaining dust could cause the anchors to fit improperly, and thus, fail.
Insert the anchor into the hole. Some anchors require that you drill holes that have either a larger or a smaller diameter than the anchor. To keep it simple, consider using sleeve anchors, strike anchors or wedge anchors -- these kinds of anchors should be inserted into holes that have the same diameter as they do.
Replace the handrail so its screw holes are aligned with the anchor holes. Turn the screws or bolts that come with the anchors into the anchors until they fit snugly. Don't over-tighten these or they could break.