For most people, shapeshifting may trigger out images of werewolves and cat-people, perhaps because these were most popularised through various stories and movies. While you can shapeshift into anything, throughout history people had most often shapeshifted into different animal forms. The purpose of shapeshifting has never been impressing others, but rather partaking of strength, wisdom or power pertaining to the specific animal and for the sake of gaining some learnings.
All shapeshifting occurs on an energy level. However, depending on the density of the vibration of a specific energy field, the shapeshifting event may be more or less obvious to physical senses. When a complete physical transformation occurs, a shapeshifting is said to be physical. Vast majority of shapeshifters do not undergo a complete physical transformation and may outwardly appear pretty much the same i.e. they will still look pretty much as any human being, and even though they may be some more or less obvious behavioral changes, other people will be able to sense the change.
According to the degree of transformation, shapeshifters are generally divided into two categories: energy shapeshifters and physical shapeshifters. With some, the ability to shapeshift has occurred spontaneously and they may have had a challenge learning how to control it. Others may have invested some time in developing this ability.
To a certain degree we are all shapeshifting all the time because our thoughts and emotions affect our energy field - the energy we project and the energy others perceive. Sometimes we do this intentionally for a specific purpose and sometimes because something affected us greatly and we lost it (just think of a time someone you know has undergone such a transformation that you'd swear he or she is an animal, and the best place place to leave such a person would be at a zoo).
If you look at how your body changes over the years and if you look at people around you - a physical change will be obvious. Not only that people age, but also that you can read their life-story from their bodies, as their bodies have over the years shapeshifted in accord with the thoughts and emotions they held dear. You may even know people who have lived together long enough that they have started to physically resemble each other as their energies synchronized. You may also know people who have over the years began to resemble in some way their pets.
Different styles of kung fu martial arts have been developed based on observation of movements of different animals. To the best of his ability, a martial artist becomes inwardly (on an energy level) an animal whose movements and particular skills he desires to emulate. He doesn't just imitate the postures outwardly, but identifies with the essence of the animal he desires to express.
Your physiology will, on the other hand, also affect your neurological response. For example, standing like a gorilla will tend to activate your reptilian brain and will create an optimal state if you have to engage in physical combat.
In Scandinavian countries, there was a special group of fierce Viking warriors - berserkers. They went into battle "without coats of mail and acted like mad dogs and wolves" (Snorri Sturluson. Heimskringla: History of the Kings of Norway. trans. Lee M. Holander. Austin: U of Texas P. 1964. p.10). The berserker was often said to change into a bestial form, or at least to assume the ferocious qualities of the wolf or bear.
Kveldulfr in Egils Saga Skallagrimsonar was spoken of as a shapechanger (Hermann Palsson and Paul Edwards, trans. Egil's Saga. NY: Penguin. 1976. p. 21), and Hrolf's Saga tells of the hero Bjarki, who takes on the shape of a bear in battle:
"Men saw that a great bear went before King Hrolf's men, keeping always near the king. He slew more men with his forepaws than any five of the king's champions. Blades and weapons glanced off him, and he brought down both men and horses in King Hjorvard's forces, and everything which came in his path he crushed to death with his teeth, so that panic and terror swept through King Hjorvard's army..." (Gwyn Jones. Eirik the Red and Other Icelandic Sagas. NY: Oxford U.P. 1961. p. 313).
Another quality possessed by berserkers is immunity to weapons. The berserker sometimes inherently possessed this immunity, while other times performed spells (that's hypnosis) to induce it. Some berserkers also had special powers to blunt weapons by his gaze. Many tales say of their berserkers, "no weapon could bite them" or "iron could not bite into him." This concept of immunity may have evolved from the berserker's rage, during which the berserk might receive wounds, but due to his state of frenzy take no note of them until the madness passed from him. A warrior who continued fighting while bearing mortal wounds would surely have been a terrifying opponent. On the other hand, if you're familiar with "iron shirt" and "iron body" practices from kung fu, you may want to compare the development of such immunity. You can also compare the development of such immunity with the practices from other traditions (e.g. yogis, dervishes, etc.)
The purpose of shapeshifting is, of course, not restricted to physical combat. In daily life you can benefit even more from using shapeshifting to gain internal strength, wisdom and many other qualities which would have a healing effect on your mind, emotions and body - the ultimate purpose being to experience greater inner balance rather than to wrack havoc.
You may already feel a particular affinity to a specific animal. For example, a cat-loving people would most likely feel affinity for shapeshifting into any kind of "big cats", while dog-loving people would most likely feel affinity for shapeshifting into "wolf-like creatures".
Shapeshifting is not restricted to animal forms, not to any creatures existing in the physical realm. Depending on your needs and desire for learning, you may choose different God-forms. In esoteric circles, this is more often referred to as "God-form assumption", rather than shapeshifting. The purpose here is to obtain learnings and to help you develop qualities and attributes within you which are characteristic of a specific deity. You may also experiment with shapeshifting into fantasy creatures, if they possess attributes you'd like to experience and/on enhance within yourself.