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I know a little about what is a turing-machine and a turing-complete language, but to understand better, could someone give examples of languages that are not Turing complete? (maybe even machines that are not Turing, as well?)
Regular expressions, in the formal definition, consisting only of:
can only recognise regular languages. A Turing-complete programming language can recognise recursively-enumerable languages.
An example is that regular expressions cannot tell you if a string consists of matched pairs of parentheses: eg
()(())is accepted while()((())()is rejected, while Turing-complete programming languages can.(Note that regexes in modern programming languages are more powerful than the formal academic definition of regular expressions. Some may even be Turing complete.)