Cognate
Cognates are words that have a common origin.
Examples of cognates are the words night (English), nacht (German), noc (Czech), and nox (Latin), all meaning night and all deriving from a common Indo-European origin. Hebrew shalom and Arabic salaam are also cognates deriving from a common Semitic root.
Cognates need not come from different languages. For example, English ward and guard are cognate.
Cognates may often be less easily recognised than the above examples and authorities sometimes differ in their interpretations of the evidence. The English word milk is clearly a cognate of German Milch and of Russian moloko. On the other hand, French lait, Spanish leche (both meaning "milk") are less obviously cognates of Greek galaktos (genitive form of gala, milk).
Cognates dish (English) and Tisch (table, German), or starve (English) and sterben (die, German), or head (English) and chef (chief, head, French) serve as examples as to how words may diverge in meaning as languages develop separately.
One may also find cognates that mean similar things, but through processes of linguistic change no longer resemble each other phonetically: cow and beef both derive from the same Indo-European root, cow having developed through the Germanic language family while beef has arrived in English from the Italo-Romance family descent.
Cognates may thus also arise through borrowings into languages. So the resemblance between English to pay and French payer originates through English borrowing to pay from Norman which, like French, had derived its word from Gallo-Romance.
Another example is French venir and Latin venire (both meaning "to come"). These words are cognates since French venir evolved from Vulgar Latin.
False cognates
False cognates are words that are commonly thought to be related whereas linguistic examination reveals that they are unrelated. Thus, for example, many people think that the Latin verb habere and Germanic haben are cognates. However, judging by the way both languages evolve Indo-European roots, the real cognate of the Germanic haben is Latin capere, "to capture" (note however that Germanic haben and English to have are cognates, and so are Latin capere and English to capture).
It has been calculated that if one takes a word from a language, there's a 40% chance that one will find a word with roughly similar sound and meaning in another random, non-related language (this is a highly doubtful statement, provide some sources/references or remove it). Because of that, even finding several hundred similar-sounding words in a couple of languages is not enough to demonstrate that the languages are related to each other. Moreover, over the course of hundreds and thousands of years, words may change their sounding completely. Thus, for example, English five and Sanskrit pança are cognates, while English over and Hebrew a'var are not, and neither are English dog and Mbabaram dog.