A heuristic is something "providing aid in the direction of the solution of a problem but otherwise unjustified or incapable of justification." So heuristic arguments are used to show what we might later attempt to prove, or what we might expect to find in a computer run. They are, at best, educated guesses.
For example, suppose one wondered if there are infinitely many Fermat primes Fn. We might argue the answer should be no heuristically as follows.
By the prime number theorem we can guess that the probability of a "random" number n being prime is at most A/log n for some real number A. So summing
A/log Fn < A/(2n log 2)
over the non-negative integers we see we should expect at most A/log 2 Fermat primes, a finite number!"
In the above naive heuristic argument several unjustified assumptions were made. For example, we assumed that the Fermat numbers behave enough like "random numbers" to make the above argument. However, Fermat numbers have special divisibility properties (see fermat divisors).