I wonder if there is any statistical test to "test" the significance of a bimodal distribution. I mean, How much my data meets the bimodal distribution or not? If so, is there any test in the R program?
Another possible approach to this issue is to think about what might be going on behind the scenes that is generating the data you see. That is, you can think in terms of a mixture model, for example, a Gaussian mixture model. For instance, you might believe that your data are drawn from either a single normal population, or from a mixture of two normal distributions (in some proportion), with differing means and variances. Of course, you don't have to believe that there are only one or two, nor do you have to believe that the populations from which the data are drawn need to be normal.
There are (at least) two R packages that allow you to estimate mixture models. One package is flexmix, and another is mclust. Having estimated two candidate models, I believe it may be possible to conduct a likelihood ratio test. Alternatively, you could use the parametric bootstrap cross-fitting method (pdf).
As mentioned in comments, the Wikipedia page on 'Bimodal distribution' lists eight tests for multimodality against unimodality and supplies references for seven of them.
There are at least some in R. For example:
diptestimplements Hartigan's dip test.
stampdata in the
bootstrappackage was used in Efron and Tibshirani's Introduction to the Bootstrap (the book on which the package is based) to do an example relating to bootstrapping on the number of modes; if you have access to the book you might be able to use that approach.
Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap.
Chapman and Hall, New York, London.
There's a question on CV that talks about identifying (i.e., estimating rather than testing) the number of modes which @whuber's search turns up. It's worth reading the answers there. One of the responses there (mine, as it happens) has a link to a Google search which turns up this paper by David Donoho on constructing one-sided CIs for the number of modes, which can of course be used as a test (e.g., if the one-sided interval doesn't include the unimodal case, you can reject unimodality). To the best of my knowledge that isn't one of the tests that Wikipedia mentions. I don't think there's an R implementation of that interval, but (in spite of the fact that Donoho tends to use fairly sophisticated tools in his discussion of it) it's actually a pretty simple idea to implement. That idea is directly related to the notion of using kernel density estimation.