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There are many problems for which it can be proven that they are, at least in the general case, unsolvable - in other words, no algorithm can ever be found to solve them. As an example, one of the first problems for which this unsolvability has been proven is the Halting Problem - determining whether an arbitrary computer program will ever stop, or run forever. Alan Turing showed that solving the halting problem is not possible, in the general case. Note that for some specific computer programs, it might be possible to show that they stop, or that they will continue running forever - but such an algorithm is not possible for ALL computer programs.

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Q: What is the algorithmic unsolvability?
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