According to Wikipedia, apophenia is:

the experience of seeing patterns or connections in random or meaningless data. The term was coined in 1958 by Klaus Conrad, who defined it as the “unmotivated seeing of connections” accompanied by a “specific experience of an abnormal meaningfulness”.

Robert Todd Carroll elaborates in Skepdic:

In statistics, apophenia is called a Type I error, seeing patterns where none, in fact, exist. It is highly probable that the apparent significance of many unusual experiences and phenomena are due to apophenia, e.g., ghosts and hauntings, EVP, numerology, the Bible code, anomalous cognition, ganzfeld “hits”, most forms ofdivination, the prophecies of Nostradamus, remote viewing, and a host of other paranormal and supernatural experiences and phenomena.

Carroll has characterized Jung’s concept of syncronicity, where unrelated phenomena and mere concidence are supposed to be of wondrous portent, as “but an expression of apophenia.”