I see that foreigners often have this problem with using correct pronouns as well as subjects :)
As you know, in English there's only one way to talk about yourself, it's "I" (or "me" or "myself", isn't it? But in Vietnamese, we have many ways. "Tôi" or "mình" is only one of them"
You use "tôi" in a formal way to talk about yourself when you're talking to other people who are not close to you or in a formal occasion which needs to be solemn or serious such as a meeting, giving a speech,..
For example:
A boss is talking to her employees "Tôi muốn các bạn làm xong bản báo cáo này trước chiều nay" (I hope you guys can finish this report by this evening)
About "mình", you use it for those who you're close (and they're at your age of course) in an informal way
For example:
Lan speaks to Minh: "Mình muốn ăn kem" (I'd like to eat an ice cream) (Because Minh is her friend so she can use this word in this situation
Or another example:
You talk to your friends: "Chúng mình nên nộp bài sớm" (We'd better submit the assignment early" (here you can use "mình" to express yourself or a group of yourself and your friends as well)
So finally, you should use "Tôi" to talk about yourself with audiences in a public place, if you use "mình", it will make you become impolite and weird perhaps :)
And one more thing: "mình" and "một mình" are completely different. As Thuy said, "một mình" means alone (without anybody else) but it's not only to talk about yourself, it's also used to talk about anybody who is doing things alone.
For example:
Cô ấy ăn cơm một mình (She is eating alone)
Mình làm bài tập một mình (I did homework by myself)
But : Cô ấy và mình ăn tối với nhau (she and I are having dinner together) (here you're talking to your friend about your date with a girl)
So you see, "mình" and "một mình" are different