The so-called 'strong form' of this hypothesis is that one cannot think a thought which cannot be expressed in one's language. The weaker form states that it is easier to think thoughts which are easily expressible in one's language, and conversely, harder to think thoughts which are not easily expressed.
Whether or not one believes this hypothesis is correct is probably strongly correlated with one's beliefs about whether language is innate or learned.
What is the Sapir-Whorf hypothesis?
The Sapir-Whorf hypothesis states that the structure of a language constrains thought in that language, and constrains and influences the culture that uses it. In other words, if concepts or structural patterns are difficult to express in a language, the society and culture using the language will tend to avoid them. The Sapir-Whorf hypothesis is considered important, and controversial to some as it can be used as a sociological argument to justify or to oppose racism and sexism (and a variety of other 'isms'). For example, the assertion that since genderless expressions in English use 'masculine' forms, English is 'sexist', presumes the Sapir-Whorf hypothesis is true.
What evidence is there to support this hypothesis?
It is known generally that people's ideas and thought change somewhat when they learn a foreign language. It is not known whether this change is due to exposure to a different culture or even just getting outside of ones own culture. It is also not known how much (if any) of the change is due to the nature of the language, as opposed to the cultural associations.
What is the current state of the Saphir-Whorf hypothesis?
The Sapir-Whorf hypothesis was important in linguistics in the 1950's, but interest fell off partially because properly testing it was so difficult as separating language from culture is so difficult. Using a culture-independant language such as Lojban is a new approach testing the hypothesis, but so far nothing has been proven.