The Third Amendment has, interestingly, never been directly litigated in a Supreme Court case hearing. It's one of the least controversial Amendments, but that's not to say it hasn't been at the forefront of some court decisions.
There have been instances where the Third Amendment has been debated in the lower courts, usually in terms of police or the National Guard. However, I'm going to focus on one of the more widespread uses.
Probably the most influential case the Third Amendment has been a part of is Griswold v. Connecticut (1965). A law in CT, passed by the predominantly Catholic legislature, banned purchasing and using contraceptives. Ms. Griswold and a medical professional who supported the use of contraceptives made a clinic to advise and prescribe their use for married couples. It is thought they established the clinic with the intention of challenging the 19th century statute.
The Third Amendment doesn't SEEM to apply here, but the majority opinion of the decision is where this comes in. This is the Warren court, possibly the most liberal Supreme Court in modern times. The Court found in favor of Griswold, in a 7-2 decision. This gets interesting when Justice Douglas wrote the majority opinion instead of Chief Justice Warren, who joined one of 3 concurring opinions. Justice Douglas describes the "penumbra" theory as his reasoning. The idea is that certain amendments, such as the 1st, 3rd, 4th, 5th, and 8th contain penumbras. A penumbra is basically a shady grey area. Justice Douglas' reasoning went on to say that the right to privacy is implied through some of these amendments. Now, it's nowhere explicitly listed in the Constitution, but using other rulings that have emanated from these amendments, Douglas' argument is that they've created this virtual right to privacy by drawing boudaries on the government's power. The concurring opinions for the most part mention the Ninth and Fourteenth Amendments. The dissenting opinion of Justice Stewart said that he thought the law was "silly", the Constitution doesn't provide a reason to strike it down. The extent of the right to privacy has been debated intensely ever since, with examples such as Roe v. Wade and the Patriot Act being some of the most famous examples.
So, while the Third Amendment has been mainly forgotten about in terms of the other amendments, it has played an important role in legislating from the bench. Many people feel it is unnecessary as it was created to help allay Anti-Federalist fears of a strong government and in direct response to the British Quartering Act. Others use it as justification or to point at the Founding Fathers' intentions of the Bill of Rights rather than the letter of the law. It's more common to see the latter used by liberal courts. Regardless, when put with other amendments, the Third Amendment plays a role in our lives today.
As an aside, every nominated Justice is usually asked about Roe v. Wade by the Senate. The Justices always deflect, whether it deals with current litigation or not, because if they divulge their true opinion, the other side of the aisle won't cofirm their appointment. As most of the Justices won't answer, they are often asked about Griswold v. Connecticut and whether it was legislated correctly. It might be worth your while to check out videos of those hearings.
Another quick fun fact is that, in 1973, the Court decided Roe v. Wade, using that right to privacy as part of their reasoning.
While r/captfaramir gives a good summary of when the Amendment has been incorporated into the decision making process, it doesn't really explain why the Amendment exists in the first place except for a brief mention of the English Quartering Act.
First thing to remember is, the Founding Fathers weren't sitting around in a room just riffing and making things up. The Constitution, and the Amendments, were very much meant to address the realities of the authors' and signers' lives.
A fundamental tenant of Feudalism was the obligation of a vassal to support his liege lord. If said Lord was passing through your lands, it was the vassal's obligation to put up the Lord and his retinue and supply them. This could be hard enough if it was just a hunting party, but imagine if the Lord was fighting a war and was leading an army.
This set the precedent so that even post Feudalism, it wasn't uncommon for European armies fighting wars in Europe to use the homes of the wealthy as make shift HQ's and board. This continued in the British colonies. The quartering of officers and soldiers in the homes of the colonists is well documented throughout the French-Indian war, the Revolutionary War and even the War of 1812.
So, this was a fundamental concept and Amendment for the fledgling nation. It was another sign that the Federal Government would be limited.
Now, it hasn't been as important as feared for two main reasons. First, initially there was great resistance to a standing federal army. No army, no need to quarter. More importantly, there have only really been two wars fought on American Soil (not including the Revolutionary War which predates this Amendment) those being the War of 1812 and the Civil War. Two wars in 200 plus years has minimized the need for such practices.