An assumption - at least in a philosophical context - is typically a presupposition, often unstated. One person makes a statement and another objects that for this to be plausible something else is being presupposed. Philosophers are often good at identifying assumptions made by other people - it comes with the territory :)
Generally speaking, it is better to have fewer or weaker assumptions, because there is less for an opponent to object to. An assumption might be weaker because it carries a smaller ontological commitment, or it avoids entailing a disputed proposition or relationship, or because it is just simpler.
A major part of philosophy is concerned with the issue of identifying and scrutinising the assumptions behind statements - at least with philosophy in the analytic tradition. If there were an answer to your question, "how do I know how many assumptions are there..." it would imply the existence of a mechanical process for doing philosophical analysis.
In practice, the process of analysis is far from mechanical. Often we only realise that there is a problem with a statement when we discover later that it leads to a falsehood, or a conflict or a confusion. Sometimes this discovery comes from new data, sometimes from thought experiments, sometimes from trying to resolve paradoxes, sometimes from trying to reconcile insights and theories from different branches of knowledge. Often, even terms and ideas that we thought were simple and unproblematic turn out to require significant analysis in order to bring clarity to a problem.
But we can't tell in advance how much analysis we'll need or what assumptions we will find in future. These are simply part and parcel of the process of expanding our knowledge and understanding of the world.