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 as are shirt and skirt.
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 may not have the same meaning: 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.
False cognates
False cognates are words that are commonly thought to be related (have a common origin) whereas linguistic examination reveals that they are unrelated. Thus, for example, many people think that the Latin verb habere and German haben are cognates. However, judging by the way both languages evolve Indo-European roots, the real cognate of the German haben is Latin capere, "to seize, grasp, capture" (note however that German haben and English to have are cognates, and so are Latin capere and English to capture).
The similarity of words between languages is not enough to demonstrate that the words are related to each other. Moreover, over the course of hundreds and thousands of years, words may change their sound 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.
False friends are similar to false cognates, but differ in that false friends, though they have different meanings, may in fact be cognates. For example, the Spanish word "embarazada" means "pregnant", but is based on "embarazar", which can also mean "obstruct" or "hamper" - the original meaning of "embarrass" ([1] - note too that the history of "embarrass" is traced through the Spanish word). Likewise, Spanish "actualmente" and French "actuellement" (which translate to "currently") derive from the Latin word actualis, and are thus cognates with the English word "actually".