Avalokitêśvara is considered to be the embodiment of the Buddhist virtue of compassion. He/she is interpreted to be the power of the Buddha Amitâbha manifested as a bodhisattva, and is therefore often depicted as the helper of the Buddha of the Pure Land 淨土. Since Avalokitêśvara is capable of manifesting thirty-three forms of incarnation 三十三身 to save people according to their capacities, he/she is depicted iconographically in thirty-three different ways 三十三觀音, which are distinguished by the number of heads and arms as well as by the attributes held in the hands. One of the most important locus classicus for the description of Avalokitêśvara is the chapter on the Universal Gate of Avalokitêśvara Bodhisattva 觀世音菩薩普門品 in the Lotus Sutra 法華經, where the rationale for his/her naming is given. 〔法華經, T 262.9.56a1〕 .

Regarding the difference in the Chinese renderings: When Xuanzang rendered the name as 觀自在, he was reading the original Sanskrit as avalokita ( 'observe' ) + iśvara ( 'unimpeded' ), which accords with the Tibetan rendering of sPyan ras gzigs dbang phyug. Kumārajīvaʼs rendering interprets the original name to be Avalokitêśvara 觀音, meaning a compassionate buddha or bodhisattva who is sensitive to, and responds to the suffering of sentient beings. Cf. 觀音信仰.

References:

Somekawa, Eisuke, Michihiko Komine, Tenyū Koyama, Hisao Takahashi, and Takayuki Hirosawa. 1993. Mandara zuten 曼荼羅図典 . Tokyo:  Daihōrinkaku.

Yü, Chün-fang. 2001. Kuan-yin: the Chinese transformation of Avalokiteśvara. New York:  Columbia University Press.

In the early iconography Avalokitêśvara is represented in the male form. He bears the head of a horse, sometimes a horse-head crown, eleven heads, a thousand eyes and a thousand arms. He dwells on a mountain of small white flowers—the symbol of purity. He holds the Cintāmaṇi—the gem with which he removes the sorrows of the sentient beings and saves them from saṃsāra. All beings are his own innermost nature and hence all sufferings are his sufferings. The prominent feature of Avalokitêśvara is compassion. He personifies compassion. He attends to the cries of the distressed. He is the protector of the afflicted. He assumes thirty forms, all being manifestations of the transformed body except the first, that is, the Buddha.