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Maxwell’s equations represent the knowledge base for electromagnetic science, on which many fields such as: electric circuits, electric components, optics, electromagnetic waves and communication technologies are all based. These equations are basically a structured combination of the classical electric and magnetic laws formed by André-Marie Ampère , Michael Faraday , and Carl Friedrich Gauss , but with a very important extension to Ampere’s Law; Maxwell’s extension to Ampere’s Law, i.e. the displacement current. In 1864, the Scottish physicist James Clerk Maxwell published four partial differential equations that describe the sources of electric and magnetic fields as well as fluxes. They also provide a clear explaination of the relation between magnetism and electricity, and how they are always conjugated as one phenomena, not two seperated things as it was thought before.
Based on this fact, the exisitance of electromagnetic waves were predicted. 
In 1887, The german physicist Heinrich Hertz solved the electromagnetic wave equation, and generated and detected electromagnetic waves in his lab. The attached pdf contains a brief explaination of these four equations (in Arabic), it is recommended to have this paper always with you when facing electromagnetic problems! (for this reason all the four equations can be found on one page in the 2nd PDF below).
Maxwell’s Equations.pdf