Think about لَيْسَ as a verb "to be not", in Arabic there is no verb "to be" but there is "to be not"
For لا it is used to give the negation to a verb, wether to give a general idea about something that one doesn't do, e.g : لا يذهَبُ الى السُّوقِ he doesn't go to the market, so we have an idea that he generally don't go to the market. Also, لا is used with a present verb. It can be used with the same construction (لا + present verb) to give an order to "not do" : لا تَذْهَب الى السُّوق = Don't go to the market.
As for لَيْسَ , it's already a verb, so it needs to give the negation to a noun / adjective.