They're all different forms of the same verb. かかる is the non-past, affirmative, plain form of the verb. This is the most basic form of the verb and is the one that all other forms are based on. It's also the one that words are listed in the dictionary as, so you'll also often hear it called the "dictionary form." In conversation, you would use this in more informal speech.
かかります is the non-past, affirmative, polite conjugation. This is the form you would use in normal, polite converstaion.
かかって is in a special form called the te-form. It has LOTS of different uses. One is that it's used to denote a sequence of actions ("I did this, and then this, and then that.") It's also combined with lots of other verbs to express more shades of meaning. The progressing form of a verb ("am doing...") is made by the te-form + いる or います. Another very common verb to combine it with is ください ("please give me") to form polite requests ("please do...for me"). And there are lots more uses of the te-form.
maxiewawa wrote:...the second with everyone else.
That's a very broad and dangerous generalization. You should only use the plain form with people who are (1) notably younger than you, or (2) you have a close enough relationship with the person.
