In a cross-sectional design, the exposure and outcome are assessed at the same time and less powerful than cohort studies or when case-control studies done with rigorous methodology.
Although cross-sectional and longitudinal studies are different research designs, this difference is not what is meant by "design effect". In survey research the term "design effect" generally refers to a measure of the extent to which the expected sampling error in a survey departs from the sampling error that can be expected under simple random sampling.
For example, if respondents are sampled in clusters (e.g. students within a class) it may be that the responses within a cluster are more similar within than between clusters, resulting in an intra-class correlation (ICC) greater 0. The larger the ICC, the larger the design effect. This design effect reduces the effective sample size to be used to calculate the standard error of an estimate (which is larger the larger the design effect). In the extreme case of responses within a cluster being the same but different between clusters, the ICC will be 1 and the nominal number of cases will be reduced to the number of clusters in the total sample. A similar problem occurs in longitudinal designs where you could treat repeated measures as being "clustered" within individual respondents. But as explained above, design effects can occur in cross-sectional studies, as well (and also in case-control studies if subjects are clustered in groups).
To account for design effects you will need special survey analysis software (for example Stata's survey commands or the "complex samples" module of SPSS). A selection of accessible references introducing the concept of design effects in survey research:
I concur with the response provided by Dirk Enzmann . I haven't looked at the references he provided, may be they address also what I am going to add here...
Two more things that matter are your outcome and the context and they are actually related.
You will probably need some informations on your outcome i.e what is known on this particular outcome in the study setting, that is the extent to which individuals belonging to a same group tend to be similar on the outcome results...
Here is a paper that you might find informative as well
In simple random sampling, the assumption is that every selection or sample is independent of each other. That means the selection of one person does not need increase or decrease of chances of selection of another individual. However, in cross-sectional study design it might happen due to various reasons that this assumption is not true leading to sampling error. In such a case study design effect may be used to increase the sample size which will give the same effect when assumption was true.
I have tried to put it in simplest manner after avoiding the technical jargon.