Disculpe: It may mean "I'm sorry" but not just as an apologize. For example, if you say "sorry for disturbing you" you should use the verb "disculpar" ("disculpe" is a formal way of this verb). So , it means, you're not asking for someone to forgive you because you did something bad but you're asking that person to give you a little bit of his/hers time. If you are asking someone to get aside so you can get into a place, for example, you may aslo use this. It is not about forgiveness, but about politeness.
Lo siento: It means you are sad, or you understand someone's sadness. For example: "I'm sorry it turned out like that" (siento que haya salido así) It doesn't mean it is your fault, but you feel someone's feelings. You can also use it as "I'm sorry", in this situation it would be the same as "perdón"
Perdón: It is anoun, it may mean forgiveness ("el perdón"), and it aslo means "I'm sorry". Maybe this one is the most used.
They can be interchangeably, but just in specific situations. As you can see they all have, as one of their meanins, "I'm sorry".
Hope thiswill make sence to you :)