Why Law of Supply indicate direct relation between price and supply? According to Wikipedia, the Law of Supply states that:


  an increase in price results in an increase in quantity supplied.


However, if the market was overflowing with a particular good (meaning higher supply), wouldn't the prices be lower? In the same vein, if the price was higher, wouldn't it be because the supply was less (all other factors, including Demand, constant). Also, wouldn't this create a run-away effect where increasing prices and supply drive more increase in each other, effectively launching prices to near infinity?

Wikipedia also states, if I understand correctly, that this view belongs to a "modern mainstream economics". 

Can someone explain this seeming paradox? When we talk about supply, are we talking about future supply which will be guided based on speculations on current prices? If so, what laws will govern the present supply?
You are confusing producer behavior and equilibrium.

If producers can sell at a price $p$ some of them will do so. Other producers cannot do so profitably at this price so the do not. At a price $p'&gt;p$ more producers can produce profitably including everyone who was already selling at $p$ so total supply increases. In this story the price was given exogenously. It was simply stated that the price is $p$ or $p'$ and we did not look at the reasons behind this. We simply stated what producers would do when facing certain price levels.

If given a price $p$ supply is larger than demand $p$ is not an equilibrium price. (An exception: it can be, if $p=0$. Would you like to buy some air from me?) You can say that this price does not occur, or at least not for a long time. As you state prices will probably decrease. In this story we looked at whether a price was actually likely given the market conditions (demand and supply). We did not simply assess what each side would do facing a given price but we tried to see if the actions of the two sides constituted an equilibrium.