'Venona' was the code name for US decrypts made on Soviet traffic, some of which were encoded using a One Time Pad. The codename in the UK was 'Bride' If used correctly, a One Time Pad is unbreakable. The Soviets knew this, and used One Time Pads for much of their encryptions, for trade and diplomatic messages as well as covert traffic.
So, how was it possible for the messages to be broken?
The Soviets had a difficulty, and that was key distribution. They needed to generate and distribute a large amount of random data. This data had to be kept secure as it was distributed to their embassies and agents. This was a monumental task.
They reused some keys on different 'channels' of communication, hoping that nobody would notice. Meredith Gardner was able to combine ciphertexts and remove the effect of the randomising key, in much the same way that the German Lorenz machine was first analysed. This allows the cryptanalysts to guess at letters on one message which used a particular key and try to find guesses which 'made sense' in the other message.
Not all of the traffic was encoded with One Time Pads, though. Over time. Using a variety of methods ranging from defections to buggings and burglaries, more Soviet traffic was decrypted.
The programme ceased in 1980, having started in the 40s. In the 1990s, Venona decrypts were released to the public in several batches and are available for download.
Note that there are, at this time, HTML errors in some links on the Venona webpages. Some links refer to things like: http://localhost/venona/venon00026.cfm
To fix this, just copy the link, paste it into the address bar, and replace 'localhost' with www.nsa.gov, so the above becomes: http://www.nsa.gov/venona/venon00026.cfm