"Hardship brings ease." المشقة تجلب التيسير
The implication of this principle is that because of difficulty, one may be given permission to do a forbidden thing or to be exempt from an obligatory one. For example, under extreme force, a Muslim is allowed to claim he's not Muslim in order to save his life. He's allowed to eat pork or drink wine if that's the only thing that could save his life. However, instead of indulging in such things, he should rather consume the bare minimum quantity necessary. As for obligatory things, permission could be given to either substitute an easier alternative (e.g., a sick person praying while sitting instead of standing) or to delay the act (e.g., delaying hajj) or to be totally exempt (e.g., women not participating in battles), all depending on the case at hand. Also depending on the situation, the permission could be either mandatory (e.g., in a life or death situation) or recommended (e.g., shortening the salah while traveling) or not recommended (e.g., combining salahs while traveling).
We find many supporting evidences for this principle in both the Qur'an and the sunnah. The Qur'an tells us that Allah intends ease for us, and not hardship. He wants to lighten our difficulties for us, and He does not charge a soul except within its capacity. The prophet urged his companions to make things easy. He never ordered them to do anything beyond their ability. When given the choice between two things, he would choose the easier one as long as it's permissible.
One should not avoid normal difficulty related to worship (e.g., waking up for fajr prayer) in the name of this principle. Doing so would contradict the Qur'an and sunnah. It's also disliked to always run after permissions.
An important exception of this principle is that it does not give permission to commit murder or adultery.