The definition of Turing-complete can be found in terms of computability. A system can be Turing-complete when it is computationally universal and, therefore, can stimulate any Turing machine.
Understanding the term
A machine could be Turing-complete if it can solve any possible computational problem by itself, provided that it is given the necessary time, memory, and instructions to do so. Many crypto enthusiasts believe that the Ethereum platform was developed using a Turing-complete language (unlike Bitcoin), but that is debatable. The co-founder of Ethereum, Vitalik Buterin, admitted many times that Turing-completeness is universal for computability but that decidability is needed for the safety of users.
We can say that a Turing-complete system, in simple terms, would be quite expressive and capable of stimulating a computer by computing any complex problem (given the time and resources). Bitcoin is not Turing-complete, given the fact that it runs on a simple consensus mechanism, which is responsible for distributing money. Solidity, the programming language used for writing Ethereum-based smart contracts, is a Turing-complete example.
Within the crypto world, Ethereum is considered to be Turing-complete. This statement implies that one could write any program to solve any computational problem of any level of complexity using Ethereum.