Democracy - government is ruled by the whims of the majority, or rather just the majority that vote. In this form of government the majority could vote however they feel at the moment even voting, for example, that all people with brown hair must pay 10% more taxes than anyone else.
Republic - government rule by law, rather than a majority. This form of government is what gives us the essence of trial by jury, where guilt is determined by comparison with law rather than a will of the majority in a given circumstance.
We also tend to think of government separated in terms of the "left" and the "right", a line of thought that is completely inaccurate. It is, rather, that government is represented with a measurement on one side as having 100% government (such as communism, fascism, dictatorship and others), and on the other side having 0% government (anarchy). The founding fathers of the United States struggled with this to create a government that sat in the 'balanced center', as W. Cleon Skousen wrote and explained beautifully in his book The 5000 Year Leap. This is the difference between 'Rulers Law' and 'Peoples Law', and his explanation of this is extremely well written.
Our US Constitution also states in Article IV, section 4: "The United States shall guarantee to every state in this union a republican form of government[...]"
This video below explains this in easy-to-understand terms, especially for those of us who need a visual representation of what we are reading in order to fully process an idea:
