From Wikipedia on 10/4/12:
A problem is an obstacle, impediment, difficulty or challenge, or any situation that invites resolution; the resolution of which is recognized as a solution or contribution toward a known purpose or goal. A problem implies a desired outcome coupled with an apparent deficiency, doubt or inconsistency that prevents the outcome from taking place. It is a component of philosophy.
Examples
Mathematical problem is a question about mathematical objects and structures that may require a distinct answer or explanation or proof. Examples include word problems at school level or deeper problems such as shading a map with only four colours.
In society, a problem can refer to particular social issues, which if solved would yield social benefits, such as increased harmony or productivity, and conversely diminished hostility and disruption.
See also: Wicked problem
In business and engineering, a problem is a difference between actual conditions and those that are required or desired. Often, the causes of a problem are not known, in which case root cause analysis is employed to find the causes and identify corrective actions.
In chess, a problem is a puzzle set by somebody using chess pieces on a chess board, for others to get instruction or intellectual satisfaction from determining the solution.
In academic discourse a problem is a challenge to an assumption, an apparent conflict that requires synthesis and reconciliation. It is a normal part of systematic thinking, the address of which adds to or detracts from the veracity of a conclusion or idea.
An optimization problem is finding the best solution from all feasible solutions. A good example of this type of problem is the travelling salesperson problem which is based on calculating the most efficient route between many places