A democratic republic is another name for a representative democracy. The United States, Japan, and most countries in Europe fit this bill. As opposed to direct democracy, which has only ever existed once, in Ancient Athens, and which requires the entire populace to participate in all decision making on an almost daily basis, representative democracies (republics) delegate the daily details of governance to officials elected by the populace at large.
In a democratic republic, or representative democracy, the will of the people gets expressed through elections, and those representatives that get elected are entrusted by the voters to govern in a way that is consistent with the wants, needs and ideals of that representatives' constituents. This system is far more practical than direct democracy, for by delegating daily governance to elected representatives and a meritocratic body of technocrats, the general population can go about its daily business, thus creating a vibrant, diversified economy.
Ironically, the most repressive regimes in the world often call themselves democratic republics, meaning that they actually put the words "democratic republic" in the name of their country, whereas as legitimate republics mostly do not. Whenever you read about a country called "The Democratic Republic of...." be wary. The Democratic Republic of the Congo, for example, is rife with corruption, state sanctioned violence against civilians, and is in almost perpetual civil war. Likewise, North Korea, the most totalitarian state in the world, calls itself the "Democratic Republic of Korea," whereas South Korea, which is actually is a democratic republic, is simply called the Republic of Korea.
The best way to find out if a country is actually a democratic republic is to see if it holds free and fair elections, where the results are monitored and opposition parties are free to campaign without limitations or intimidation. A real democratic republic will often have numerous parties vying for power, and perhaps most importantly, it will have a history of peaceful transitions of power between rival parties.
Finally, true democratic republics have civilian oversight of the military. This last part is crucial because even legitimate democratic republics are not immune from military coups, and the ability of a republic to keep its military separate from its political leadership is crucial to maintaining the integrity of both. That is why in many democratic republics, such as the United States, members of the military are forbidden from publicly expressing their political preferences.