Quick answer
nunca = never, the neutral everyday word.
jamás = never, stronger and more emphatic, close to "never ever".
Both follow the same grammar: before the verb they stand alone, after it they need no. Together, nunca jamás is the strongest form.
Side by side
Nunca voy al gimnasio. / Jamás iría al gimnasio.A plain habit versus a flat refusal.
No lo hago nunca. / Nunca lo hago.After the verb you need no; before it you do not. Same rule for jamás.
Nunca jamás.The two stacked together are the strongest "never" Spanish has.
Memory trick
Default to nunca. Save jamás for the moments where English would say "never ever" or "not in a million years".
Typical mistakes
Jamás desayuno en casa, prefiero el bar.
Nunca desayuno en casa, prefiero el bar.
A daily habit is neutral, so it takes nunca. Jamás turns breakfast into a solemn vow.
Lo he visto nunca.
No lo he visto nunca.
When the negative word comes after the verb, Spanish still needs no in front. The double negative is obligatory.
Check yourself
Fill in: "___ he probado el sushi, pero quiero."