Question

what is the meaning of this sentence?

the president can never claim the direct popular mandate that a prime mnister can.this ensures that she remains a nominal executive.

Answer 1

This sentence means, that our country has a prime ministerial rule, rather than a presidential rule. That means the prime minister has all the authority, over the government, over the state and is responsible for decision making. But, on the other hand, the president executes no such power. The president is just a ceremonial head you may say, he has no power over that state whatsoever. 
You can compare this to government in England (correct me if I am wrong), where the Queen is just someone who they will show their head, but she doesn't executes any power. All the power is with the prime minister, they make the decisions for the country. The same thing is in our country.

Answer 2

Answer:

See, mandate means the authority, or the power to make decision, or control something. So direct popular mandate means, that the person has direct controls over a large no. of people (a whole country in this case).