A problem is anything you'd like to be better. Anything to improve. It could be about how well you understand something, or it could be a practical problem like your foot is caught in some mud.
It's not bad to have problems. Problems alone don't make us unhappy. What makes us unhappy is having problems and then not being able to make progress towards a solution. It's getting stuck and failing to solve problems that is unpleasant.
The way to solve a problem is to create knowledge of the solution. That may sound like a truism, but it offers a useful perspective: it reminds us to use what we know about creating knowledge. Problem solving follows the standard approach to knowledge creation: imaginative conjectures and criticism.
Problems are inevitable. There is no lifestyle that avoids all problems. But there are lifestyles that have less problems or smaller problems. Problems are also soluble: there's always something we can do about them.
Although all problems have solutions, they may not have the type of solution you initially want. Sometimes we have mistaken goals, and the solution involves changing our mind about what to want. Whenever this is the case, there will always be some good reason, and some explanation of how this change is entirely in our best interest. If a new mindset is better, there must be some way to voluntarily and happily change to it.
When a problem involves two or more people who disagree, then a solution requires a new idea about how to proceed that everyone is happy with. That means everyone has to be open to changing their existing preferences to better ones (better by their own standards). If people's initial ideas conflict, as they often do, then logic dictates that some people will have to change their mind in some way for them all to agree on a solution. Typically, for any complex issue, since nobody is perfect, everyone will change at least a little.
Science and other fields of knowledge are often seen as a history of solutions. First we invented theory X, then theory Y, and so on. They can equally well be seen as a history of problems. First people worked on solving problem P. Then the solution led to a new problem, Q, and so on.
When we judge ideas, we should think about what problem they are trying to solve and whether they solve it. If you don't know what problem an idea is supposed to address — if you don't know its purpose — then how can you evaluate it?
Acknowledgments: some ideas presented are modified from, or inspired by, ideas from Karl Popper, Ayn Rand, William Godwin, Edmund Burke, David Deutsch and Thomas Szasz.