Taijiquan, or Tai Chi Chuan, is an internal martial art originating from or around Chen Jia Gou ('Chen Village') in Henan province in central China roughly 500 years ago (although its precise origins are a matter of ongoing debate); without doubt, however, the Chen family are the progenitors of the original art, although they themselves are deeply indebted to countless generations of Daoist monks, mystics, and misfits who preceded them. The traditional spelling according to the Wade-Giles system, Tai Chi Chuan, dating from over 100 years ago, is quite different from the modern spelling according to the Pinyin system developed by Chinese linguists in the 1950s, which is in widespread use today - note however that there is no difference between Tai Chi Chuan and Taijiquan except that the latter spelling is more universal.
Taijiquan is a martial art first and foremost, but the paradox is that because it is an extremely powerful one, it takes years of training, attention, and discipline to develop the power necessary to cope with an unscripted martial situation. Therefore, it is vital that a thorough groundwork in posture, relaxation, alignment and balance has been established before even considering the martial applications of the art, and indeed the very purpose for which Taijiquan is intended! Once more the hallmarks of Daoism, the ancient Chinese philosophy of Yin and Yang as laid out in the Dao De Jing (Tao Te Ching), written by Laozi (Lao Tzu) over two thousand years ago, shine through.
Precisely because Taiji is based on Daoist principles, it is a complete system in and of itself, and does not require - nor indeed preclude - supplementation with other forms of exercise; you need no equipment to practise, and only a small space is necessary. You can train wherever you like, alone or in a group, as much or as little as desired. One thing, however, is certain: it will profoundly restructure not only your body, but your mind and soul as well, since it is a holistic regime which works on all levels of the human organism. Postural tensions are linked to psychological states, and by releasing these tensions you also release mental energy, insight, and clarity. As you relax the body, you relax the mind; as you relax the mind, you relax the body. As the Dao De Jing admonishes, we should seek to "Empty the mind, and fill the belly; weaken your ambition, and strengthen the bones."
Qigong (spelt 'Chi Kung' in the Wade-Giles system), written 氣功 in Mandarin, is an ancient Daoist exercise system based on very similar principles to Taiji (emphasising gentleness, balance, coordination, light stretching, using the whole body as one, moving from the centre, and so on), except that Qigong is more static in nature, with no (explicit!) emphasis on martial training. The core exercise is Standing Qigong, or Zhan Zhuang (站桩), which teaches posture and alignment through a standing meditation exercise which can last from a few minutes to an hour or more, as desired, but never beyond your natural limits! As such, embodied mindfulness is a key element of Qigong training, and this aspect of the Taiji curriculum is unique to Chen Style Taijiquan. In Mandarin, 氣 'qi' means 'breath or 'energy' and 功 'gong' means 'work'; Qigong therefore means 'breath work' or 'energy work', depending on your interpretation. Note that the 氣, 'qi' in Qigong (Wade-Giles: Chi Kung) has no relation whatsoever with the 极, 'ji' in Taijiquan (Wade-Giles: Tai Chi Chuan).