A pair of false cognates consists of two words in different languages that appear to be or are sometimes considered cognates when they're really not. Note that there could be an indirect connection between them, however only words sharing a common root can be considered real cognates.
For example, the word for "dog" in the Australian Aboriginal language Mbabaram happens to be dog, although there is no common ancestor or other connection between that language and English (the Mbabaram word evolved regularly from a protolinguistic form guduga). Similarly, the Korean word manhi (an adverb meaning "plentifully") resembles the English "many".
False cognates are particularly common in core or common vocabulary, as shown by this last example. Such cognates often occur in kinship terms (!Kung ba and French papa (both "father") or Navajo ma~, Chinese ma1 and English "mother") or in numbers (Korean tu and English "two").
Although perhaps not technically accurate, the term "false cognate" is sometimes used to describe a false friend. The difference between a false cognate and a false friend is that while a false cognate means roughly the same thing in both languages, a false friend generally means either the opposite (Welsh ie = "yes" vs. Japanese iie = "no") or something completely unrelated.