Don't play with any online casino.  Take some time to check them out and avoid getting burnt.

There are online casinos who will simply cheat you out of your money.  They may simply not payout winnings or they may use software which is unfairly biased to favour the House.  Obviously you want to stay clear of these operations.  But how?  First, make sure they're licensed and that the license is from a reputable country (see below).  Secondly, check they are members of a watchdog organisation.  Reputable casinos will publish their payout figures and accounts from a reputable accountancy firm (like Price-Waterhouse-Coopers.)  Check the casinos website for these things.

You may also like to check out some gambling message boards for any players who are unhappy with an online casino or who recommend one.

Online Players Association
Online Gaming Council

Online casinos operate and are licensed from different countries.  Most are outside the USA, although if you're American you can play them and receive payouts.  However, some countries are less reputable than others.  There have been a number of scams reported that operated from a few non reputable countries while other countries have built a reputation for honesty.   

Countries that offer legit casinos that you can confidently play with are...

United Kingdom (including the Isle of Mann and Gibraltar)
Ireland
Australia
Curacau - Dutch Antilles
Antigua
Barbuda

The UK, Ireland and Australia are developed first world countries with strong government infrastructure and enforced regulations on gambling.  These countries have strict laws and regulatory bodies that monitor and probe casinos for ethical standards and practices.  The other countries listed are not as developed but are serious about gambling enforcement and licenses.

888 Casino, Labrokes, Players Only and Superbook Casino are all reputable online casinos.

Countries to avoid are...

Dominica
Honduras
Bahamas
Nicaragua
Honduras
St Kitts
Madagascar
Haiti
St Vincent
Aruba
St Martin
Any East European Country

These countries don't have the infrastructure or money to regulate gambling.  The licenses in these countries are also very cheap allowing just about anyone to set up for business.  Many of the scams reported by unhappy gamblers operated from these countries.