Both the size of the animal at time of tagging as well as its potential future size is a major consideration in the type of tag that should be applied to a fish. For large coastal sharks (sanbar, tiger, blacktip), we almost exclusively use Casey M-dart tags (which are distributed by the NMFS Apex Predator investigation program) These tags have the steel 'M' type tip that fits into the tip of a tagging pole, with a short lentgth of monofilament line attached to the tip that leads to a capsule containing a piece of waterproof paper with the tag number and return information. These tags are applied just below the base of the dorsal fin into the body musculature.
We use a long (12 ft) tagging pole when tagging animals in the water that are too large to boat and a short (4 ft) tagging pole to tag medium size fish that are taken into the boat. It is easiest to apply these tags with a pole because they are most easily applied with a sharp thrust into the target area of the animal. A well placed thrust will penetrate the skin and fully seat the tag in the musculature and connective tissue beneath the dorsal fin.
The tags are secured to the pole by inserting the tag head into the tagging needle, and placing a rubberband over the tagging pole to hold the trailing end of the tag (the rubberband comes off or breaks when the tag is applied). A similar type of tag, which may be the type to which the author of the question is referring, has the M type tip with a length of yellow plastic tubing which has the tag number and return info written directly on the tubing.
These type of tags should be applied with the same method as described above. Both of these types of tags do well in large fish, and also have enough length to them so that they cannot be overgrown by the animal so that they totally disappear inside the body. However, these types of tags are too large to apply to neonate large coastal sharks, and small coastal sharks (3ft TL or less) without major harm to the animal. In the case of neonate large coastals (which have a lot of growing to do) we use Rototags, which are applied by punching a small hole in the upper half of the dorsal fin with a leather punch and clipping the tag together thru the hole.
These tags fit somewhat loosely in small sharks, and thus allow for substantial widening of the dorsal fin as the animal grows, and if the tag does become overgrown, the affected area is in a fin and not in the body cavity. In small coastal sharks (sharpnose, blacknose, bonnethead, etc.), we almost exclusively use plastic dart (aka Hallprint) tags, because these animals do not grow to as large a size and thus will not overgrow these smaller tags (Hallprints are available in a variety of sizes as well).
Effective application of plastic dart type tags is much trickier however, and the 'right' method to apply them will vary with personal opinion, but this is how I do it: A tagging needle is used to apply the tag, the entire body of the tag slides down into a large needle, with only the head exposed, this needle is inserted on the left sid of the dorsal fin just below the posterior margin of the dorsal fin at an angle parallel to the axis of the body and at a slightly inward angle.
The needle is run forward under the dorsal fin until the head of the tag is near the suface of the skin on the OPPOSITE ANTERIOR margin of the dorsal fin. The idea behind applying the tag under the dorsal fin is to insert the tag amidst the many hard cartilage rays that support the dorsal fin, giving the tag a good area in which to firmly anchor.
Another aspect adding to the difficulty in applying these tags is that the single plastic anchoring barb found at the tip of these tags tends to snap of when attempting to pierce some of the thicker skinned sharks (like blacknose), so a 'pilot' hole thru the skin is sometimes necessary (the tip of a sharp knife can be used, but be careful not to puncture past the skin and injure the animal).