To keep things simple let's assume you have a perfect random number generator (i.e. I will discuss only the statistics not the numerics of the problem). I will also focus on the practical matter and gloss over some mathematical details.
From a practical perspective "convergence" means that you will never get an exact answer from Monte-Carlo but increasingly good approximations. Try out your 100'000 paths example. The two values for the price of your option will be slightly different everytime you use a fresh, i.e. independent, sample.
Two mathematical theorems are relevant to describe convergence: First, the law of large numbers, which says that the average of independent samples converges to the expected value (i.e. price) and the central limit theorem, which tells you that the distribution of the error converges to a properly scaled normal distribution. This justifies what Mark Joshi is alluding to in his post.
You mention a typical and very relevant question: What size samples do I need to achieve a certain prescribed accuracy? If you assume normal distribution of errors you can calculate a confidence interval and solve this expression for the sample size. You will often hear people say that Monte-Carlo "converges very slowly" or "converges with $\sqrt{n}$". This is because to achieve a tenfold increase in accuracy you need a hundredfold increase in number of paths. For a serious study of this important topic I recommend the book by Paul Glasserman