Dan and seidel's answers are great. Here a few more general thoughts on the topic.
To a certain degree the definition or description of replicates may be relative and depend on the question you are asking. However, one way to think of this issue is as follows. A purpose of replicates is to estimate or account for the variability in a measurement or observation. These estimates speak to the reproducibility and ultimately the precision of an experiment. It doesn't really matter what you call your replicates, the key is the think about the source of the variability you seek to estimate by performing your replicates. Vigilantly thinking about the nature and sources of variability is a critical step towards developing sound experimental designs. Also it is never too early to think about what statistical approaches/tests you are going apply in evaluating your replicates. Form hypotheses about the sources of variability and their relative magnitude and contribution to your overall variability and seek to test those hypotheses.
Ask yourself, is the variability I expect to encounter across my replicates related to a technical or biological process? For example, in the experiment you describe you are using multiple microarrays. The manufacturing of these arrays is not an exact process. Replicates where the only variable changing is the microarray slide being used are going to be considered technical in nature. Similarly, scanning of the microarrays is not perfectly reproducible and introduces variance of a technical nature. Replicates that seek to estimate the variability associated with anything related to manufacturing, instrumentation (e.g. variability in pipetting volumes when you add your siRNA reagent), reagent batches, etc. related to the 'technological' aspects of your method can likely be classed as 'technical replicates'. You may have some grey areas where a biological process (e.g. enzymes) are involved in the creation of a reagent. Experiments that involve tissue culture, model organisms, etc. are going to have both technical and biological sources of variation. For example, RNA expression patterns measured from the cells in your cultures will vary according to cell cycle of cells in the culture, mutations acquired during culture, and possibly many other biological processes related to cell growth, etc. Replicates that seek to measure variability associated with such factors can likely be considered 'biological replicates'. If you seek to demonstrate that an observation is general to any cell type, you might consider repeating the experiment in different cell lines to also be 'biological replicates'. However, as mentioned in the comments, this may introduce so many sources of biological variability that it is almost of a different experiment.
Finally, as has already been noted, a replicate may involve an overlap of both technical and biological sources of variability. Biological replicates in particular are usually influenced by both technical and biological processes. The generic term 'experimental replicate' may be used to describe a replicate of the entire experiment comprising a complicated mix of multiple sources of variability.
All answers are good, anyone interested on the question should read them all (three at the moment I am writing).
h.mon. If you like all three answers you can actually "accept" (i.e., put a check mark next to) them all. Up to you. Just thought I'd mention in case you didn't realize this.