A corner solution is a special solution to an agent's maximization problem in which the quantity of one of the arguments in the maximized function is zero. In non-technical terms, a corner solution is when the chooser is either unwilling or unable to make a tradeoff.
In economics when someone says "I wouldn't buy that at any price" or "I will do X no matter the cost", those are corner solutions. Another example is "zero-tolerance" policies, or parents who are unwilling to expose their children to any risk, no matter how small and no matter what the benefits of the activity might be. "Nothing is more important than my child's safety" is a corner solution in its refusal to admit there might be tradeoffs. The term "corner solution" is sometimes used by economists in a more colloquial fashion to refer to these sorts of situations. The word "corner" refers to the fact that if one graphs the maximization problem, the optimal point will occur at the "corner" created by the budget constraint and one axis.
A corner solution is an instance where the "best" solution (i.e. maximizing profit, or utility, or whatever value is sought) is achieved based not on the market-efficient maximization of related quantities, but rather based on brute-force boundary conditions. Such a solution lacks mathematical elegance, and most examples are characterized by externally forced conditions (such as "variables x and y cannot be negative") that put the actual local extrema outside the permitted values.
Another technical way to state it is that a corner solution is a solution to a minimization or maximization problem where the non-corner solution is infeasible, that is, not in the domain. Instead, the solution is a corner solution on an axis where either x or y is equal to zero. For instance, from the example above in economics, if the maximal utility of two goods is achieved when the quantity of goods x and y are (−2, 5), and the utility is subject to the constraint x and y are greater than or equal to 0 (one cannot consume a negative quantity of goods) as is usually the case, then the actual solution to the problem would be a corner solution where x = 0.
In consumer theory
The more usual solution will lie in the non-zero interior at the point of tangency between the objective function and the constraint. For example, in consumer theory the objective function is the indifference-curve map (the utility function) of the consumer. The budget line is the constraint. In the usual case, constrained utility is maximized on the budget constraint with strictly positive quantities consumed of both goods. For a corner solution, however, utility is maximized at a point on one axis where the budget constraint intersects the highest attainable indifference curve at zero consumption for one good with all income used for the other good. Furthermore, a range of lower prices for the good with initial zero consumption may leave quantity demanded unchanged at zero, rather than increasing it as in the more usual case.