What is Finite Element Analysis?

What it does

For me here, applying FEA in its simplest form, it calculates the stresses and elastic deformations across the entirety of structures / components under load.
That's any component

of any shape (wow!)
under any simultaneous combination of loadings
of any material or combination of materials in one structure
etc

That was an engineering "holy grail", before the development of FEA to far more complex tasks somewhat eclipsed the perception of significance of this initial achievement.

For a more general answer, see eg Wikipedia page - "Finite element method".

"Approximate solution" - what does that mean?

In "my" narrow case of stresses and deformations plus the general case of FEA, FEA like other computational-numerical methods give "approximate" solutions.

How "approximate" is "approximate"? Not very... That is mainly a theoretical point. Because there is no "exact" solution for most structures, you cannot compare the FEA solution to an exact solution. Couldn't you have a matching combination of a physical structure and a FEA model of it and compare what the two give, to resolve this once-and-for-all? That wouldn't work. The problem is the random variabilities in the real physical world affecting any physical structure are bigger than the "approximation" in the FEA solution.

That's also true for any structure you make for a service.

Therefore, for all practical purposes, the answers obtained from a "converged" FEA model of a structure can be stated as the stresses and deformations the structure has.

Some widely-stated general statements about computer-numerical solutions

A general point made about computer numerical solutions: an approximate solution across all variables is much more valuable than exact solution across only some variables.

A limitation of computer-numerical solution is that the relative influence of different variables is not revealed by the computational output.

This contrasts with a mathematical expression. Taking for example the equation expressing Ohm's Law for electrical resistance, "V=IR"; it can be seen within the equation that the current increases in direct proportion to the voltage. Eg if you double the voltage you double the current.

The way this is dealt with is that you have computer-numerical solutions to predict quantities across complex systems, while the underlying science provides overall knowledge of how the variables behave and interact. Knowing both, the engineer knows what to look out for in designing something where the physical situation is complex.

(R. Smith, 03Jun2016)