The difference between un and l'un is fairly simple - it has to do with grammar and register.
1) When un is a pronoun,* rather than an article or number, it can be replaced by l'un in formal French:
- Tu dois choisir l'un de ces livres
You have to choose one of these books
J'ai vu l'un de ses amis
I saw one of his friends
2) In addition, when un is a pronoun at the beginning of a sentence, it is nearly always replaced by l'un, for reasons of euphony:
- L'un de mes meilleurs étudiants est à l'hôpital
One of my best students is in the hospital
L'un de vous doit m'aider
One of you has to help me
- c'est tout l'un tout l'autre - there's no in between, everything is black and white
- de deux choses l'une - there are two possibilities
- l'un à l'autre - to each other
- l'un après l'autre - one after the other
- l'un dans l'autre - all in all
- l'un d'eux, l'un d'entre eux, l'une d'elles, l'une d'entre elles - one of them
- l'un et l'autre - both (of them)
- l'un l'autre - one another, each other
- l'un ou l'autre - either one, one or the other
- ni l'un ni l'autre - neither one
- soit l'un soit l'autre - either one, one or the other
J'ai un frère
I have one brother
Je vois un homme
I see a man

