How to select and use Strong Magnet together with hall sensor and amplifier?
The most popular types of magnets are Neodymium-Iron-Boron (sintered or bonded), Samarium-Cobalt, ferrite (often bonded), Aluminium-Nickle-Cobalt (often called Alnico), and ceramic magnets.  They all have different properties in terms of magnetic field strength, resistance to high and/or low temperatures, resistance to demagnetization, cost, etc.
The magnet in the photo is most likely a sintered Neodymium-Iron-Boron (NdFeB) magnet with a Nickel-Copper-Nickel coating to help protected it from corrosion.  NdFeB magnets are produced in a variety of different strengths and temperature grades.  This will be called out as a number and a letter, like 35UH or 50M.  The number tells you the energy product of the magnet (which gives you a rough idea of the strength of it) and the letters tell you what the Intrinsic Coercivity of the magnet is (which gives you a rough idea of the maximum temperature the magnet can be used at).  
The equivalent for a magnet is called flux density and it is measured in Tesla, Gauss or sometimes "lines per square inch."  Flux density is a measure of flux per unit area.  "Typical" magnetic field strength at a certain distance is going to depend on the specific grade of NdFeB and the thickness of the magnets.  Once you have those, you can find calculators online that will give you the answer.  
Once you have a material picked out, the main things you need to know are shape (square, rectangle, circle, etc), area and thickness of the magnet.  
Sintered NdFeB magnets are made in the following way:  Raw materials are melted down and then formed into a fine powder.  The powder is pressed into a shape (this can be done in a couple different ways -axial pressing or isostatic pressing).  Then they are sintered (heated).  Then they can be and often are machined, which could be slicing a bigger block into smaller parts or grinding in order to meet a certain tolerance.  Then they are coated to prevent corrosion.  The coating could be an epoxy, Nickel-Copper-Nickel, or some other method.  Once this is done, they are magnetized by being placed in a strong magnetic field.