Order of operations - why are they in the order they're in? I understand the order of operations, but why are they ordered the way they're ordered? Is there a particular reason why multiplication should have a higher precedence than subtraction, other than to prevent ambiguity?

Edit: I'm a curious software developer that's relatively lousy at math. A simple explanation that your grandma could understand would be very welcome. :-)
The precedence rules allow efficient representation of polynomial expressions in a concise normal expanded form. If you reversed the precedence of multiplication then the notation would be more efficient for factored polynomial expressions. But, generally, polynomial operations are much less efficient in factored form (compare, e.g. equality testing and addition). 

The reason that polynomial rings are ubiquitous is their universality (they are free $\rm\:R$-algebras). So, for example, any polynomial identity that you prove true in $\rm\:\mathbb Z[x]\:$ will hold true over any ring, e.g. the binomial theorem.
To illustrate your example with substraction and multiplication, denote them as two variable functions: $S(a,b)=a-b,\ P(a,b)=a\cdot b$. Then to calculate $a-b\cdot c$(without any brackets), we would have $S(a,P(b,c))$. In order to calculate the result, we need to find $P(b,c)$ first. 
There are many operations defined with the aid of the simpler ones: multiplication is defined using addition; powers are defined using multiplications. It is then natural to give priority to the elevated operations, since they depend on the simpler ones.
Think practical. In a warehouse there are 2000 bottles of Coca cola, in boxes of 20 pcs. Someone takes out 30 boxes. How many bottles are left? Answer: $2000-30\times 20$. If we calculate $2000-600=1400$ we get the practical result right(we go and count the remaining bottles to see that... ) . If we calculate substraction first we get $1970\times 30&gt;2000$, and we see that something is wrong.


I think the third aspect is the most important, since mathematics is made to help modelling different things we encounter in real life. Not all maths is abstract. If you take the operations and change their priorities, you will get results which do not correspond with reality.