Which word to use when we ask for help?
- We know that the person asked is able to do it.
- We don't know if the person asked is able to do it.
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As waiwai993 answered, Can/May/Will have different meanings. However asking someone if they can help you usually implies that you would like their help, and it gives the person an opportunity to decline without being rude. Maybe they are able to help in theory, but can't right now because they are too busy. In that case they can say that they can't help. If they simply don't want to help they can just decline without a reason. So I would always use "Can you help me with this" or "Could you help me with this" unless I needed a more specific case (such as I KNOW that they CAN, but I am forcing to answer whether they WILL or not, or I know that they are ABLE, but maybe their mother won't let them*).
* Note: I think it's very unusual to use "May you help me" and would probably never say that under any circumstances, unless I was trying to be ironic.
I would actually offer a couple additional options:
Under condition 1 (that you know that the person is able to help, but you're unsure if they will):
Would you help me with this?
Under condition 2 (you're unsure if that person is able to help):
Could you help me with this?
These are similar in meaning to will and can respectively, but are a more polite way of asking. I would especially recommend these if you're in more formal setting, talking to strangers, etc.
Can you help me with this?
Technically, this asks if it is physically possible for the other person to help. However, it is commonly used to mean "Will you help me with this?"
May you help me with this?
This is equivalent to "Are you permitted to help me with this?"
Will you help me with this?
This is the one you probably want—it basically asks whether or not the other person is going to help.
As for me, I sometimes feel a bit shy talking to people, so I prefer polite variants like:
Can you please help me with...?
Can I ask you for help with...?
Or you can just say:
I need your help.
I don't have much experience in communication with English-speakers, but I think those variants can be used in different situations.
Why not ask etymology for some help here?
The word can comes from Proto-Germanic kunnan
to be mentally able, to have learnt
Can is about ability, skills, knowing how to do things, whereas the word may comes from Proto-Germanic root mag-, infinitive maganan, from Proto-Indo-European magh-
to be able, have power
May is about power, the position a person is in to grant or refuse another person permission to do something, says the Online Etymology dictionary.
Can I go to the bathroom?
– etymologically – is not really appropriate for asking permission as it means something like Do you think I know how to go to the bathroom?, which might sound rude, even! On the other hand,
May I go to the bathroom?
– etymologically again – means something like Would your highness allow me to go to the bathroom? and, in the classless world we live in (?) we are not really ready to your-highness anyone if we can avoid it! Well, little children cannot really avoid it for a question of stature and status, but, when you have grown up, using may is felt as humiliating, as if you were kneeling in front of someone.
Could I go to the bathroom?
represents a middle ground between rudeness and self-humiliation, which can satisfy both the speaker and the listener!
Using the modal auxiliary can in the conditional, 'unrealising' the present, making it hypothetical, dampens the shock. I am asking about objective conditions (do you think I know how to) but using the conditional instead of the present puts more emphasis on the listener, on their ability to assess the situation, which is a tacit acknowledgement of their authority.