'I' and 'me' are different cases of the personal pronoun.
Simply put, 'I' is used as the subject of a sentence, 'me' as the object.
Examples:
- I am a good swimmer. (well this is an easy one, we don't say 'Me am...')
- He kicked me. ('me' is the object of the sentence and thus we don't say 'He kicked I')
'Myself' is not a substitute to 'me' or 'I', but it is used to give emphasis, as in:
- She doesn't like it, but I myself find it very attractive. (we want to emphasize the difference between her opinion and my own)
'Myself' can be a substitute to 'me' when referring to an action taken about the speaker's person, like:
- I want to hit myself in the head. (we don't really say 'I want to hit me...')
As I am not a language teacher, those of you who are please comment or correct my remarks if wrong in any way!
Simply put, 'I' is used as the subject of a sentence, 'me' as the object.
Examples:
- I am a good swimmer. (well this is an easy one, we don't say 'Me am...')
- He kicked me. ('me' is the object of the sentence and thus we don't say 'He kicked I')
'Myself' is not a substitute to 'me' or 'I', but it is used to give emphasis, as in:
- She doesn't like it, but I myself find it very attractive. (we want to emphasize the difference between her opinion and my own)
'Myself' can be a substitute to 'me' when referring to an action taken about the speaker's person, like:
- I want to hit myself in the head. (we don't really say 'I want to hit me...')
As I am not a language teacher, those of you who are please comment or correct my remarks if wrong in any way!

Another explanation:
All three are personal pronouns. "I" is the subjective form, used when the speaker is the subject of the sentence as in "I am a good swimmer", or when the speaker is a predicate nominative, as in "The best swimmer is I."
Use "me" as a direct object ("He kicked me"), as an indirect object ("He gave me the book"), or as the object of a preposition ("He gave the book to me").
"Myself" is the reflexive form, used when the subject and object are the same, as in "I want to hit myself on the head." It is also common, as dinosm indicates, to hear "myself" used to emphasize. Its use, however, should be limited to instances where the speaker genuinely intends to emphasize his or her importance. Like the word "very", it is overused.
All three are personal pronouns. "I" is the subjective form, used when the speaker is the subject of the sentence as in "I am a good swimmer", or when the speaker is a predicate nominative, as in "The best swimmer is I."
Use "me" as a direct object ("He kicked me"), as an indirect object ("He gave me the book"), or as the object of a preposition ("He gave the book to me").
"Myself" is the reflexive form, used when the subject and object are the same, as in "I want to hit myself on the head." It is also common, as dinosm indicates, to hear "myself" used to emphasize. Its use, however, should be limited to instances where the speaker genuinely intends to emphasize his or her importance. Like the word "very", it is overused.
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Comments
"I hurt myself." or "I shot myself in the foot."
The same thing goes with 'himself.' For example:
"He hurt himself." or "He shot himself in the foot."
Saying, "She doesn't like it, but I myself..." is not proper.
Trying even harder to avoid the lowly me, many people will substitute myself as in 'The suspect uttered epithets at Officer O'Leary and myself.' Myself is no better than I as an object. Myself is not a sort of all-purpose intensive form of me or I . Use myself only when you have used I earlier in the same sentence
Are either of the two correct? If so, which one? I am writing a letter of resignation. Thanks!
You need this:
On behalf of my family and me, we ....
CJ
I am the owner of that book. I want to read that book.
Use "me" when it is the object:
That book belongs to me. Hand that book to me.
Use "myself" to modify the subject:
I wrote that book myself.