Dialect is variation between different groups of speakers of the same language. For example, American English and British English are dialects of English. There's also dialects within American English (General American, Mid-Atlantic, etc) and within British English (Received Pronunciation, Cockney, etc.) They are natural and usually can be understood by speakers of the same language (not everyone agrees on that, though).
Jargon is a set of specific words used by a specific group of people that usually can't be understood by anyone outside of that group, even within speakers of the same language. The group is usually some kind of field of study or of work. For example, the average English speaker probably wouldn't understand what an "intervocalic voiceless bilabial fricative" is-- that's linguistics jargon. Also encountered a lot with computers; most speakers wouldn't understand what a "random-access memory fault" is, either.