To lose one's head - is it a metaphor or metonymy?
Head here probably stands for the life of a person,so it's probably a metonymy?
And is it the same for phrase to give smb. a heart ?
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Metaphor and metonymy are both ways of connecting two concepts (or making one concept stand for another). The fundamental distinction between metaphor and metonymy is that metaphor connects two concepts based on similarity and metonymy based on part/whole relationship (meronymy).
Typical examples of metaphor are 'Our relationship is on rocky ground' or 'We had to battle over who will use the toilet first.' You could easily make a list of point in which the two domains are similar (e.g. when you're on rocky ground, you have to step carefully and in our relationship, everything I do has to be done with care). Note, the similarities are always partial, although sometimes people will debate over where they begin and end.
Typical examples of metonymy are 'Nice wheels' where 'wheels' stand for 'car' or 'France voted no' where 'France' stands for the representative doing the actual voting.
In some cases, you can tell them apart through a simple substitution test. If you can substitute the one word or phrase that's being used to stand for something for the word or phrase used to describe it and get exactly the same meaning, you're more likely to deal with metonymy.
For example, you can change 'Nice wheels' to 'Nice car' or 'France voted no' to 'The representative for France voted no' and get exactly the same meaning. However, with metaphors like 'I gave her my heart', you cannot simply substitute 'I gave her myself'. (Of course, in many cases, you can do a substitution with metaphor as well. For example, 'All men are pigs' could be rephrased as 'All men are nasty.' But you can see that this does not convey the same meaning and would require much more elaboration than when you substitute 'car' for 'wheels'.)
So, to answer your specific question, To lose one's head is a metaphor. The phrase is intended to give an image of 'panic'. Although, 'head' often metonymically stands for people (as in '10 dollars per head') or mind/thoughts ('his head comes up with interesting ideas'), you cannot simply say that 'To lose one's head' is the same as 'To lose oneself' or 'To lose one's mind'. (They are in a similar area but have quite distinct meanings.)
The same goes for 'to give someone one's heart' (to fall in love) or 'to give someone heart' (to give someone hope or courage). It is the whole statement that does the job, and not just the 'hear'. But heart can easily be used metonymically as in 'There are a lot of brave hearts among us' (brave people) and 'Look at those young hearts in love' (young people). These metonymies also rely on the metaphorical connections between heart and courage and heart and love.
This brings us to the question raised by @jlawler of whether it is worth making the distinction between metaphor and metonymy. Some people (George Lakoff, among them) would claim that they represent a crucial distinction in figurative language. Others (myself included) would argue that they are in fact achieving a similar conceptual job and often work in conjunction. However, the stereotypical cases are quite straightforwardly different and it is worth maintaining the distinction between them.
I've described the difference between metaphor and metonymy in more detail in a longer post on MetaphorHacker.