Define "guanxi" for me. It seems to not translate so well to English, but I suspect it has something to do with one's personal network.

Thanks.

Wikipedia is a best friend.  And it actually explains guanxi quite well IMHO

Guanxi describes the basic dynamic in personalized networks of influence, and is a central idea in Chinese society. In Western media, the pinyin romanization of this Chinese word is becoming more widely used instead of the two common translations—"connections" and "relationships"—as neither of those terms sufficiently reflects the wide cultural implications that guanxi describes.[1]

Closely related concepts include that of ganqing, a measure which reflects the depth of feeling within an interpersonal relationship, renqing (人情 rénqíng), the moral obligation to maintain the relationship, and the idea of "face" (面, miàn), meaning social status, propriety, prestige, or more realistically a combination of all three.

At its most basic, guanxi describes a personal connection between two people in which one is able to prevail upon another to perform a favor or service, or be prevailed upon. The two people need not be of equal social status. Guanxi can also be used to describe a network of contacts, which an individual can call upon when something needs to be done, and through which he or she can exert influence on behalf of another. In addition, guanxi can describe a state of general understanding between two people: "he/she is aware of my wants/needs and will take them into account when deciding her/his course of future actions which concern or could concern me without any specific discussion or request".[citation needed]

Guanxi refers to the benefits gained from social connections and usually extends from extended family, school friends, workmates and members of common clubs or organizations. It is custom for Chinese people to cultivate an intricate web of guanxi relationships, which may expand in a huge number of directions, and includes lifelong relationships. Staying in contact with members of your network is not necessary to bind reciprocal obligations. Reciprocal favors are the key factor to maintaining one’s guanxi web, failure to reciprocate is considered an unforgivable offense. The more you ask of someone the more you owe them. Guanxi can perpetuate a never ending cycle of favors.[2]

The term is not generally used to describe relationships within a family, although guanxi obligations can sometimes be described in terms of an extended family. The term is also not generally used to describe relationships that fall within other well-defined societal norms (e.g. boss–worker, teacher–student, friendship). The relationships formed by guanxi are personal and not transferable.[citation needed]

When a guanxi network violates bureaucratic norms, it can lead to corruption, and guanxi can also form the basis of patron–client relations.[citation needed]

In East Asian societies the boundary between business and social lives can sometimes be ambiguous as people tend to rely heavily on their closer relations and friends. This can result in nepotism in the work force, as it is common for authoritative figures to draw from family and close ties to fill employment opportunities; instead of assessing talent and suitability such as is the norm in Western societies. This practice often prevents the most suitably qualified person being employed for the position.[3] Although guanxi is associated with the traditional Confucianist doctrine, guanxi ties were strongly developed during the Mao Zedong government (1949–1976), particularly due to the work unit system,[4][verification needed][unreliable source?] which led many workers to construct strong social networks within their units, thus improving their ability to enjoy important resources and privileges.

Douglas Guthrie distinguishes between guanxi and guanxixue (关系学, the 'art' or 'knowledge' of guanxi)[5], as the former is considered in the modern Chinese society as common inter-personal ties that reflect the Chinese nature, while the latter represents the manipulation and corruption brought about by a selfish and sometimes illegal utilization of guanxi. Although many Chinese lament the strong importance of guanxi in their culture, they still consider guanxi as a Chinese element that should not be denied.
Usage examples

Someone is described as having good guanxi if their particular network of influence could assist in the resolution of the problem currently being spoken about.

Guanxi is most often used in the Western media when interpersonal obligations take precedence over civic duties, leading to nepotism and cronyism.[6][7]

Guanxi is also a newly introduced word used anecdotally among human resources specialists and managers, for the ability to legally discriminate against people who do not belong within a favored clique of individuals, because the people being discriminated against do not belong within a clearly delinated legally protected group.
Similar concepts in other cultures

Sociologists have linked guanxi with the concept of social capital (it has been described as a Gemeinschaft value structure), and it has been exhaustively described in Western studies of Chinese economic and political behavior.[8]

SEAJ