'Democracy' refers to who has the power to compel his/her/their will. Whoever possesses the power to compel is a 'ruler.' All countries have a ruler, whether one person or a group of persons. The largest group that can rule is the People. In America, the People are supposed to have that power - they are the rulers - hence we have a Democracy. DEMOS = PEOPLE, CRATIA = POWER/RULE
'Republic' refers to a representative form of government bound by law (The Constitution). In America, those in power (the People) elect those to secure their rights and liberties and to uphold the law (The Constitution). Hence we also have a Republic. RES = "AFFAIRS/ MATTERS," PUBLICA = PUBLIC.