Is it a religious duty for a woman to have sex with her husband whenever he requests it? What if she is not in the mood? Should a man be understanding or is it expected from his wife to always obey even when she doesn't want to?
Yes that's true. But men shouldn't take it literally. There must be kindness and understanding. Islam ordered the husband to be nice and gentle and understanding. Mohammad (pbuh) said:
إن من أكمل المؤمنين إيمانا أحسنهم خلقا وألطفهم بأهله
“Indeed, the believers with the most complete faith are those with the most excellent character and are most kind to their women.” 
And in Quran:
يَا أَيُّهَا الَّذِينَ آمَنُواْ لاَ يَحِلُّ لَكُمْ أَن تَرِثُواْ النِّسَاء كَرْهًا وَلاَ تَعْضُلُوهُنَّ لِتَذْهَبُواْ بِبَعْضِ مَا آتَيْتُمُوهُنَّ إِلاَّ أَن يَأْتِينَ بِفَاحِشَةٍ مُّبَيِّنَةٍ وَعَاشِرُوهُنَّ بِالْمَعْرُوفِ فَإِن كَرِهْتُمُوهُنَّ فَعَسَى أَن تَكْرَهُواْ شَيْئًا وَيَجْعَلَ اللَّهُ فِيهِ خَيْرًا كَثِيرًا
O you who have believed, it is not lawful for you to inherit women by compulsion. And do not make difficulties for them in order to take [back] part of what you gave them unless they commit a clear immorality. And live with them in kindness. For if you dislike them - perhaps you dislike a thing and Allah makes therein much good.[4:19]
And guess what did Mohammed (pbuh) said in his Last speech to Muslims which means it is important:
ألا واستوصوا بالنساء خيراً فإنما هن عوان عندكم.
"Treat your women well. They are as captives in your possession." [Ref]
Also, Muslim men have more responsibilities towards women than women towards men. He is responsible of protecting them, providing (food, clothes, money..etc) and being kind to them. In addition to that, Muslims are strictly prohibited from oppressing any one, so what about his own wife? Also, this is something which is a priority for most men but not most women (Usually), and it affects the stability of the relationship. Obviously, the man has to judge his situation and see if the timing is mutually conducive.
The believing men and women whom Allah Subhanah has blessed to be bonded in the sacred institution of marriage, have certain rights towards each other in their marriage. And one of the biggest rights of marriage is that the couple may enjoy conjugal relations with each other to their heart's desire. Unless the wife has a legal and valid Shariah reason for not allowing the husband to have intercourse with her (menses, post child-birth, while fasting in Ramadan, etc.), there are no conditions or circumstances when either the husband or the wife may refuse the invitation of the other.
Abu Dhar al-Ghafari reported:
The Messenger of Allah (saws) said: "there is a reward for you [even] in sex with your wife."
The companions asked: "O Messenger of Allah (saws) is there a reward if one satisfies his passion?"
He (saws) said: "Do you know that if he satisfies it unlawfully he has taken a sin upon himself? Likewise, if he satisfies it lawfully, he is rewarded."
Related by Ahmad and Muslim.