Most airfoils are built using the NACA formulas/tables where each parameter is specified in the type of foil.
Each of the 4 parameters specific the foil type NACA XYZZ where
X--First digit describing maximum camber as percentage of the chord.
Y--Second digit describing the distance of maximum camber from the airfoil leading edge in tens of percents of the chord.
ZZ--Last two digits describing maximum thickness of the airfoil as percent of the chord.
A symmetrical foil would be NACA 0010 would mean 0% camber, 0% percent camber distance and 10% thickness to chord length.
There is also a 5 digit series of NACA foils for more complex shapes. And there are 16-series, 6-series, 7-series and 8-series foils as well. The benefits of using a NACA shape is they have been well characterized. Here's a summary of the types and the pros/cons depending on your application.
Edit: From an application perspective (non-academic) Foil design and analysis is usually done by defining what type of lift parameters you want: range of lift coefficients, Reynolds numbers, where the airfoil should perform best, stall characteristics, moment coefficient, thickness, low drag, high lift... This is where NACA foils are handy because you don't have to start from scratch.
On a custom foil the next step is usually to analyze it numerically with one of two popular methods:
PROFIL by Professor Richard Eppler, University of Stuttgart, Germany.
XFOIL by Professor Mark Drela, Massachusetts Institute of Technology, USA.
Take a look at how each tool works and you'll get an idea of methods that might be useful for you to learn. Other references the vortex lattice method, basic thin foil aerodynamics, thin foil aerodynamics derivations text book