Proving whether an input sequence is in a given RE language
A language is recursively enumerable if there exists a Turing machine that:


halts and says "yes" whenever its input is in the language;
either halts and says "no" or just doesn't halt whenever its input is not in the language.


The undecidability of the halting problem means that there is no algorithm which, when given an RE language (or, rather, when given the code of a Turing machine which accepts some language) and a string, can determine whether the string is in the language or not. In general, all you can do is run the Turing machine and see if it accepts its input. However, if it seems to be taking a long time to run then, in general, there's no way of figuring out if it's in an infinite loop, so even this doesn't actually work.

However, that doesn't mean that you can never tell whether a particular machine will have certain behaviour.  For example, it's easy to tell that the machine

IF the first character of the input is 0 THEN accept
ELSE [do something complicated]


will accept any string that begins with&nbsp;$0$.  More sophisticated analytic techniques will allow you to prove that more complicated machines do or do not accept certain inputs.  However, what the lack of a general algorithm means is that, whatever analytic techniques you develop, there will be some machines and some inputs (in fact, infinitely many of them) where your techniques fail. In those cases, your only options are to develop even more advanced techniques (which still won't work in all cases) or to give up and just run the machine.