The following six steps are essential to know in order to understand what the (mathematical) approach of the FEA is:
First step:
Preprocessing:
 Create geometry
 Defeature small unnecessary details in the model (\rightarrow careful since this is a source of errors)
 Make use of symmetries  not possible if the load is not symmetric
 Fillets used in the model may lead to stress singularities (“sharp reentrant corners”)
 Define material properties
 Linear material behaviour is straight forward
 Elastoplastic material (Hardening rules, behaviour beyond yielding, etc.)
 Isotropy  Anisotropy
 Choose initial and boundary conditions
 MasterSlave Assignment
 Define (if necessary) side conditions like contact definitions
 Discretization of the geometry \rightarrow Meshing of the components
 First Order & Second Order Mesh
PreAnalysis Checks:
 Consistent units (Young’s Modulus, Dimensions of the components etc.)
 Check loads and boundary conditions
 Check Mesh Quality visually (some softwares offer an automatic Mesh Quality Check)
 Check Element Types / Accuracy
 Check Contact definitions if necessary
 Check that you only write the results you need
Second step:

Element formulation \rightarrow development of equations for elements

Set up the partial differential equation in its weak form
Third step:
Assembly:
Set up global problem \rightarrow obtaining equations for the entire system from the equations for one element.
Fourth step:
Solving the equations (solve system of linear equations).
Fifth step:
Postprocessing:
Determining quantities of interest, such as stresses and strains and obtaining visualizations of the response.
Sixth step:
PostAnalysis Checks to verify results:

Check if the deformed component looks as you would expect
\rightarrow Pay attention to the scaling! 
Convergence Study (“Does a finer Mesh have significant influence on the results?”)

Compare simulation results to analytical solutions if available
For general information about errors and singularities in FEA, please have a look at our blog post:
Errors in FEA and Understanding Singularities (Beginners’ Guide)
Read more about the Finite Element Analysis in our dedicated article in the SimWiki.
Also see our SimScale Simulation Wiki for more about other interesting simulation related questions.