This question seems to come up quite a lot. When one questioner asked for comments on how similar the two languages are in comparison with English and any other language, many people suggested that Czech and Slovak are as close as two dialects of English, such as the dialects of Scotland and England.
Slovak and Czech certainly seem to be mutually intelligible. My wife has conversations with her Czech friends where she speaks Slovak and they reply in Czech, and everyone chats away as happily as if they were speaking the same language. A few years ago I watched a version of Test the Nation which went out in both Slovakia and the Czech Republic and set out to discover which nation was the smartest (the Slovaks, of course!). They had one Slovak moderatorka (presenter) - the ubiquitous Adela Banášová - and one Czech, and they simply presented it bilingually, the Slovak presenter speaking Slovak and the Czech one speaking Czech.
However, this mutual intelligibility is exaggerated because for many years before the break-up of Czechoslovakia, Czechoslovak TV (you never use the adjective "Czech" to refer to the government, institutions etc of Czechoslovakia in my home unless you want a beating - a point which in Britain is lost even on professional historians like Peter Clarke in Hope and Glory) used to broadcast in both Czech and Slovak. Apparently the language of the nightly news used to alternate between the two. So most Slovaks and Czechs above the age of thirty were regularly exposed to the other language in their childhood. The importance of this in developing an understanding of the other language is suggested by the difficulty which today's Slovak children have in understanding Czech.
Personally, I have always found Czech and Slovak quite similar, though there is a small but significant part of the vocabulary which is entirely different. It is more like the similarity between Spanish and Portuguese, though, than between different dialects of English.