Depending on your question (and which dimension of biodiversity you are considering), there are different packages in R that are very helpful. As someone already pointed out, "vegan" is good (for some stuff), but for functional diversity, I'd recommend the "FD" package which has both Rao's Q and FDis, and for phylogenetic diversity, I'd recommend the "picante" package.
I concur with Kevin, it really depends on the soecific question you want to address. I have used CANOCO (with CANODRAW) and PRIMER and found them very useful for various aspects of biodiversity analyses. However, these are not free.
I think that Estimates is a very good program, also you can find useful the program "Tree diversity analysis" , wich accompanied the book from the same name. The authors are Roeland Kindt and Richard Coe, edited by World Agroforestry Center.
PAST free biodiversity analysis is really very good. I have lot of satisfaction because its free and near about complete multivariate analysis package. I have 1 publication from past also...
For species richness estimation i recommend EstimateS (http://viceroy.eeb.uconn.edu/estimates/) for niche overlap and other analyses ECOSIM (http://www.garyentsminger.com/ecosim/index.htm) and for other ecological analyses the BIODIVERSITY R )http://cran.r-project.org/web/packages/BiodiversityR/BiodiversityR.pdf) and its text book: http://www.worldagroforestry.org/resources/databases/tree-diversity-analysis
You can download Biodiversity Pro Software here https://www.sams.ac.uk/science/outputs/
but note that the programme has not been updated for more recent operating systems it may not function fully on Windows PC operating systems beyond XP. It can run on Windows XP but may lose functionality when installed on other systems such as Windows Vista.
Maybe suggest it would be best to search the software from R. There are a list of softwares, in which you may select and download the package, I am sure you can find the answer there.
Biodiversity Pro does not take "0' or "null' values(figures) it deviates the final analysis results. PAST would be helpful in that case. OR you may play with "R" software.