Applications of design theory I have recently started reading up on design theory, with the ultimate purpose of doing some original research in that area. I understand the mathematics fairly well, but am not understanding the application. Can someone here explain that? Alternatively are there some articles, web-sites, papers explaining the application of design theory in agriculture, biological experiments etc.
For example after establishing the existence of/constructing a certain type of design mathematically, how do we use that knowledge in practice? 
 A: I'll try to give a basic idea of how these are used in exploratory medical research.
Traditional medical experiments:  Patients are given a single candidate treatment and are observed for a given response (i.e. dose-response).  If the response is statistically significant, then it is considered for further studies, and possibly may end up on the market.  Observations may necessarily take years to conduct, and suitable patients may be hard to find.
This slow process is undesirable if there is only a small chance of the candidate treatment working.  Instead we can use a scheme along the lines:


*

*We have a number of patients (or test animals) with a disease (or diseases).

*We have a large number of candidate treatments.

*We treat the patients with some subset of the possible treatments.

*We observe for a large number of responses.


Combinatorial designs are used to determine which patients receive which treatments in such a way that if a given response is observed, then the structure of the design would indicate the treatment that caused it.  We also want to minimise the cost of the experiment (so having fewer patients is better).  We don't need to deduce for sure that treatment X causes response Y, we just need to reduce the possibilities significantly; more in-depth and specialised studies can be conducted for plausible candidates.
Essentially, it's faster and cheaper to conduct exploratory research using a scheme based on a combinatorial design.  Moreover, inadvertent discoveries, such as the highly profitable Viagra, may also be made this way.
This is an example in a medical setting, but hopefully it's clear how the general idea extends to experiments throughout science.  There are other ways, such as factorial experiments, in which combinatorial designs are used.


For example after establishing the existence of/constructing a certain type of design mathematically, how do we use that knowledge in practice?

It usually doesn't work that way.  Most often, we start off with a hypothesis (or a family of hypotheses) and design an experiment around it/them.  Real-world constraints often result in mathematically inelegant questions.  Futher, understanding why a certain design is or isn't suitable may require domain-specific knowledge.
Charlie Colbourn once told an interesting anecdote at a summer school I was attending; some chemists asked him to design an experiment according to some criteria.  He gave them a design that met their criteria, but for some reason the chemists ignored his reply.  He asked some other chemists about what happened, who explained that if they mixed the chemicals according to his design, they would blow up the lab.  (Of course, this is second-hand information, but hopefully we can appreciate how domain-specific knowledge is necessary.)
A: EDIT: replaced dead link.
See this site; "The extract deals with her interest in how a simple mathematical model applies to the weaving process and the use of combinatorial designs for agricultural experiments." 
A: You can check the link given below. 
http://www.sciencearchive.org.au/scientists/interviews/p/notes_cp.html
Interviewer: It is quite serious mathematics but still something that a general audience can understand. You have been interested also in group theoretical aspects of designs.
Professor Cheryl Praeger : Yes. My interest in design theory, the theory of combinatorial designs, came about in two different ways. The first was indeed looking at the symmetry of designs, arising first from work of Peter Cameron. But I also became involved in designs used for experimental layouts for agricultural experiments that statisticians would analyse – to help statisticians to understand what symmetry groups were involved in the particular experimental designs which they were interested in. This became a collaborative work with Terry Speed, for whom it was first a teaching and then a research interest. He was trying to understand what types of designs statisticians might be interested in, to get a feeling for the class of designs that they needed to understand: he went away for a year in 1978–79, and every time he found a new design in the research literature somewhere he would send it back to me and say, ‘Analyse this one.’ The object was to analyse the variance of this design, and in the analysis we could point out and identify the various factors which were significant in trying to analyse data arising from using this layout, for example, comparing the yields of different varieties of wheat.
