Q:

C++ Expression Templates

I currently use C for numerical computations. I've heard that using C++ Expression Templates is better for scientific computing. What are C++ Expression Templates in simple terms? 

Are there books around that discuss numerical methods/computations using C++ Expression Templates?
In what way, C++ Expression Templates are better than using pure C?

A:

What are C++ Expression Templates in simple terms? 

Expression templates are a category of C++ template meta programming which delays evaluation of subexpressions until the full expression is known, so that optimizations (especially the elimination of temporaries) can be applied. 

Are there books around that discuss numerical methods/computations using C++ Expression Templates?

I believe ET's were invented by Todd Veldhuizen who published a paper on it 15 years ago. (It seems that many older links to it are dead by now, but currently here is a version of it.) Some material about it is in David Vandevoorde's and Nicolai Josuttis' C++ Templates: The Complete Guide. 

In what way, C++ Expression Templates are better than using pure C?

They allow you to write your code in an expressive high level way without losing performance. For example, 
void f(const my_array<double> a1, const my_array<double> a2) 
{ 
  my_array<double> a3 = 1.2 * a1 + a1 * a2; 
  // ..
}

can be optimized all the way down to 
for( my_array<double>::size_type idx=0; idx<a1.size(); ++idx ) 
  a3[idx] = 1.2*a1[idx] + a1[idx]*a2[idx]; 

which is faster, but harder to understand. 

A:

Adding to sbi's answer, expression templates implement high-level peephole optimizations using templates for pattern matching and synthesis.
They also add syntactic sugar, or make your code more readable, by allowing you to specify the algorithm in terms of simple operations. So, in this case, simplicity and elegance are achieved through optimization by metaprogramming. At least, if you do everything right.

A:

There is a nice article on C++ template math in the good old Flipcode archive (sure brings back memories):
http://www.flipcode.com/archives/Faster_Vector_Math_Using_Templates.shtml

