Social networks are networks in which vertices represent people or groups of people and edges are social interaction among them, e.g. friendship. In sociological terms vertices are actors and edges are ties.
The study of social networks goes back to 19th century psychiatrist Jacob Moreno who became interested in the dynamics of interactions within groups of people. Moreno  called his diagrams sociograms which later become known as social networks. In his study on schoolchildren he used triangles and circles as vertices to represent boys and girls respectively. A friendship relationship is indicated by an edge connecting two vertices. The diagram reveals that there are many friendships between two boys and two girls, but few between a boy and a girl. Once drawn the diagram, it was easy to see. This is what persuaded social scientists that there was merit in Moreno's methods.
Depending on the question one is interested in answering there are many different ways to define an edge in such an network. Edges may represent friendship, professional relationships, exchange of commodities, communication patterns, romantic or sexual relationships, or many other types of connections between individuals.
The techniques to probe different types of interaction may involve direct questioning (e.g. interviews) [Rea97, Rap61], direct observation of experimental subjects, the use of archival records (e.g. the "southern woman study" [Davis41]), ego-centered data analysis [Burt84, Bern89], affiliation analysis [Davis41, Gal85], small-world experiments [Mil67, Trav69], snowball sampling [Erick78, Frank79, Thom00], contact tracing and random walk sampling [Klov89, Thom00]. This techniques have been applied to problems like friendship and acquaintance patterns on different scales of the population, e.g. students, professionals, bord of directors, collaboration of scientists, movie actors, musicians, sexual contact networks, dating patterns, covert and criminal networks such as drug users or terrorists, historical networks, online communities and others.
A classic problem of social network analysis is to discover clustering. In the reminder of this article we will focus on different empirical methods used to measure social networks.
Asking people questions is the most common way to accumulate data about social networks. This can be done in the form of direct interviews, by questionaries or a combination of both - each with advantages and disadvantages with respect to the quality of data. A good introduction to social survey design and implementation is Rea and Parker .
Surveys typically employ a name generator - a mechanism to invite respondents to name other nodes in the network as well as their relationship to them in order to explore the network. In the study of Rapoport and Horvath a question to the schoolchildren was to name eight best friends within the school.
There are some interesting points to notice about name generators. Nominating other vertices by ties is an asymmetric process. Person A may nominate person B as friend but there is no need that person B has to nominate person A as friend. Therefore it makes sense to represent this data as directed networks. Vertexes in directed networks have two types of degree: in-degree - the number of individuals who identified the vertex as friend - and out-degree - the number of friends identified by the vertex.
The second point concerns the limit of responses given to the respondents. In the study above the limit was to name up to eight friends. Such fixed choice studies limit the out-degree of the vertices. This cut-off may lead to the loss of information about the small-world effect of the network, which is caused by a small number of highly connected vertices. However, the in-degree is not affected by such cut-offs.
Studies based on direct questions are not only laborious, inaccurate and costly. Most of all, the data contain uncontrolled biases.
For determining network structure, sosiometric studies - such as in the previous section require a survey of all or nearly all of the population. A reconstruction of the complete networks of ties is not possible. Given the high costs to survey large networks, a study of personal networks or ego-centered networks may be a feasible alternative. An ego-centered network is a network about one individual (ego) and its surrounding immediate contacts (alters).
A typical survey would be to sample the population at random and ask them to identify all those with whom they have a certain type of contact. Also, they are asked for information about characteristics of themselves and there alters. This type of of survey is useful in particular if we are interested in the degree of the network. A random sample of degrees can give a reasonable degree statistics. In case we also gather information about contacts between alters, we can also estimate clustering coefficients. If we have data on characteristics of egos and alters we can estimate assortative mixing.
Direct observation over a period of time is an obvious method to construct social networks. This is a rather labor-intensive method. It is restricted to small groups, primarily ones with face-to-face interactions in public settings. It is the only viable experimental technique for social network studies in animals.
A highly reliable source of social network data is archival records.
Affiliation networks are special kind of social networks to focus on cluster discovering.
The best general introduction on network theory is the book from Mark Newman . There is an active research community mainly somehow affiliated with the Santa Fe Institute.