Data are available publicly on several websites e.g. icpsr, databank, UK Data Archive. I am seeking guidelines, references and examples about using these open data for research. In other words, I want to publish a paper based on analyzing some publicly available datasets. Do I need permission first or they are available actually for me to use them in my research ?
A public data set is just that: public. They have been made available for the specific purpose of being used by other people.
Thus, you generally do not need permission, but you will need to appropriately cite the source of the dataset, thereby giving appropriate credit to its originators. For a well-curated collection, you can find appropriate citation information embedded within either the "packaging" or metadata of the dataset or at the online source where you obtain it.
It's hard to speak in generalities: each database will be different.
Databases normally will have some sort of usage agreement associated with them. For example, DrugBank posts the following prominently on their front page:
DrugBank is offered to the public as a freely available resource. Use and re-distribution of the data, in whole or in part, for commercial purposes (including internal use) requires a license. We ask that users who download significant portions of the database cite the DrugBank paper in any resulting publications.
Other databases may not be as up-front about the limitations. Instead, they may have a usage page accessible somewhere on their website. (Usually there's a link in the header or footer.) For example, the NCBI provides their usage disclaimer at the bottom of each page, under a Polices and Guidelines link. Likewise, the UK Data Archive provides guidance under a Terms and Conditions link at the bottom of the website.
If I had to summarize, though, public data repositories are generally fine with you publishing academic work which makes use of their data - that's why they're providing it in the first place! Conditions on publishing, when present, are generally involved with to requiring appropriate citation, or things like not trying to de-anonymize personal data (both of which you should be adhering to anyway). The non-trivial usage restrictions tend to revolve more around redistribution of the data sets, either by as-is or in processed forms.
That said, you can't make any general statements about usage - each database will have it's own terms and conditions, most of which should be easily available from the same location you obtain the data.