Tax is the largest single liability that a company is likely to experience and it is sensible to take to take steps to minimise tax liability so it's no wonder that businesses look to create an offshore company to minimise their liability.
First some terminology. Tax evasion is evading tax. This is illegal. Deliberately evading tax is most jurisdictions is illegal. Tax avoidance is not illegal. This is designing a corporate structure to minimise tax liability and complying with the laws of the jurisdictions where the company operates. This is perfectly legal. So offshore company formation in most situations is perfectly legal.
This kind of legal structure is not limited to far flung lands like Thailand. Thinking of buying a holiday home in France? Be careful. You may fall foul of French inheritance law which can easily over-rule the inheritance laws of the country in which you are domiciled and make your whole estate subject to French inheritance tax. French laws dictate who can inherit your possessions - over-ruling the wishes laid out in your will. Establishing a company to own your little piece of France can avoid the minefield of international inhertiance law.
The second reason why these jurisdictions are popular is secrecy. The laws in the jurisdiction can create secrecy since there are few (or even no) disclosure requirements.
Countries like the UK have very high levels of transparency and information about the financial state of companies are publicly available. Datalog makes lots of this data available for free.
Picking a jurisdiction with high levels of secrecy does suggest that the reason for the offshore company is tax evasion rather than tax avoidance, so be careful. World leaders are attacking jurisdictions such as Switzerland since they are seen as harboring companies and individuals who are evading tax.
UK territories such as the Channel Islands and Isle of Man are safe choices since they have very high levels of transparency (actually higher than most USA states), low taxation and also importantly - credibility.
Delaware is a popular choice since it's a low tax regime and also because the levels of transparency are low. It's possible to hide behind service companies so that information about the officers and shareholders is hidden from prying eyes.
Panama is very topical right now. Interestingly the transparency levels for Panama are very high. Although you need to register to use the Panama company register (and use Google translate to navigate the Spanish), it's possible to find the directors, shareholders and a range of documents about offshore companies. Delaware doesnt meet the same levels of disclosure as Panama. How long before the media picks up on this?
Transparency levels for the offshore UK territories is very high. Information is freely available for companies in these territories so transparency is high however there may be a charge to access certain documents. However some other states/countries don't even disclose the existance of the documents let alone provide access!
It is easier to research companies that provide offshore company formation services for less secretive jurisdictins. For example XEPHYR LIMITED provides formation services for the USA state of Delaware. Details can be found via the Xephyr website.