Turing-complete as an adjective when describing programming languages simply means that it is capable of computing any function that can be computed using a Turing machine (recursively enumerable function). Equivalently, we could also say that the programming language or system is capable of simulating or being simulated by a Turing machine. Then, the question becomes what a Turing machine actually is. It's essentially the simplest model of a general-purpose computer possible, devised by Alan Turing so as to make it possible to reason about the process of computation itself without getting bogged down in the practical details of physical computers. A simple description of a Turing machine is a tape, with an unlimited number of spaces on which symbols can be written or overwritten, a tape head that marks the current position of the machine, and a set of states where the machine may be in. At any given time, the machine is in one of its possible states, and it can do any or all of the following things:
- Stop processing
- Overwrite the symbol on the tape head with a new symbol
- Move the tape head to the left or to the right
- Go to a different state
This is the simplest possible model of fully general-purpose computation that has been devised. It has been proved that these simple things suffice to simulate every possible model of computation out there. It is enough to compute everything every computer everywhere in the world that has ever been built can compute (the question of whether it can do so efficiently is another matter of course, and that gets into details we don't consider here).
It becomes possible to prove certain interesting things about what functions can be computed by a Turing machine, in fact, it can be shown that certain functions cannot be so computed, i.e., a Turing machine trying to compute them might never stop, and there'd be know way of knowing whether it ever would, but that's another story.
Well, I took a graduate-level course in the Theory of Computation a few years back, so I think I'd know about these things.