Algorithm. An algorithm is a computational method that can be represented in a well-defined formal language within a finite amount of time and space for solving an abstract problem.
Starting with the initial state and input, an algorithm describes a computation that proceeds through a finite number of states, eventually producing an output and terminating at a final state. The simplest form of algorithms deals with decision problems, and produces yes-or-no outputs.
Per the Church-Turing thesis, any algorithm can be simulated by a model of computation known to be Turing-complete (e.g., Turing machine, λ-calculus).
If the transition of states is deterministic, the algorithm is called a deterministic algorithm; otherwise, it is called a probabilistic algorithm (or randomized algorithm, non-deterministic algorithm).
Is every problem solvable/decidable (Entscheidungsproblem)? The answer is no. Examples of undecidable problems: the halting problem and Hilbert’s tenth problem.
Can every problem that can be verified in polynomial time be decided in polynomial time? We don’t know. See the P versus NP problem.