It can run using a number of solvers, you can find a list here: http://www.minizinc.org/software.html

It's a language for specifying problems that can be solved through constraint satisfaction. I'm unsure if such languages are even Turing complete. Their main advantage are insanely succinct problem specifications. F.e the n-queens problems: https://github.com/MiniZinc/minizinc-benchmarks/blob/master/... MiniZinc solves it for boards up to 100x100 in under a second.

“Constraint Programming” refers, for the most part, to the solving of finite-domain problems via backtracking and constraint propagation and is typically concerned with performing a complete, global search of the solution space.

https://en.m.wikipedia.org/wiki/Constraint_satisfaction

I like this answer. However, by solution space do you mean domain space?

Yes, sorry, the search space.

MiniZinc is a modeling language - for problem specification. Then there are solvers that understand MiniZinc (Gecode, ECLiPSe, Google OR-Tools, ...), and they use variety of algorithms (mixed linear programming, constraint-specific algorithms, local search, SAT-solving techniques, ...). You can use the same specification of the problem in MiniZinc with different solvers.

Possibly a simplex solve for a large linear problem.

There is a lot off different type of solvers that can be used to solve a MiniZinc model: MIP, Constraint Programming, SAT, Hybrid SAT+CP, Local search etc.