Traditionally, the Bill of Rights were a group of amendments added to the Constitution so that it could be accepted by enough states to become validated.
Today, each amendment has its own unique issue with current day politics (bar the 3rd amendment)
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances."
A very relevant amendment, and a powerful one too. It means that, as long as people abide by peaceful requirements, they are allowed to pretty much say what they want. Some people abuse this amendment by claiming that it revokes the ability to disagree with someone - this is the exact opposite of what happens: freedom of speech protects people to say whatever they want, but it also allows anyone to form a rebuttal. It simply claims that government censorship is forbidden. What is interesting is that often on a smaller scale (and even on a larger scale), this particular amendment is broken, in the case of things like banned books.
Most parts of this amendment don't really face much opposition in terms of defining it (outside of libel and slander cases), but freedom of religion is an exception. Many social issues balance on this, and, because it's hard to draw the line between "equal rights" and "religious beliefs", it often pops up in modern culture.
"A well regulated militia, being necessary to the security of a free state, the right of the people to keep and bear arms, shall not be infringed."
I.E. Right to bear arms. A very vague amendment that has lead to some controversy due to the advancement of gun technology, making them much more dangerous than when this was written. Bottom line though, it is THE amendment that prevents the government from a complete militarized takeover unlike many other states - nobody likes to argue with a shotgun.
Personal opinion: While regulations on firearms are not breaking this rule (you still have the right to have them, just not go and do some stupid ♥♥♥♥ with them), outright eliminating them does go against this amendment. Depending on the future of firearms and American culture, it may be necessary to produce another amendment that revokes this one (similar to Prohibition) but it's much too early to tell.
"No soldier shall, in time of peace be quartered in any house, without the consent of the owner, nor in time of war, but in a manner to be prescribed by law."
Due to the United States never fighting on their own home turf for the last 100 years, this amendment is irrelevant. It was written as a response to Redcoats forcing American colonists to house soldiers in wars, such as in the French and Indian war.
"The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized."
This basically means that the government cannot search through your stuff without a reason. There are exceptions, such as the common plain sight (sorry kids - having a bong in your car is not covered by this amendment). This is actually the amendment that sparked all the controversy regarding the NSA and Snowden, so I believe it's an extremely relevant amendment, especially considering the computer age. However, in very few circumstances, such as child pornography, it is reasonable to excuse a person's privacy for protecting, say, children being taken advantage of and abused.
"No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a grand jury, except in cases arising in the land or naval forces, or in the militia, when in actual service in time of war or public danger; nor shall any person be subject for the same offense to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation."
Basically, when you go are arrested for a crime, you have the right to be treated fairly by the courts. You cannot be charged with something and have any damages to your person or your belongings until you are convicted.
Although not as frequent as some other amendments, I would still say this one is very relevant, as it forces courts to at least have some fairness.
"In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury of the state and district wherein the crime shall have been committed, which district shall have been previously ascertained by law, and to be informed of the nature and cause of the accusation; to be confronted with the witnesses against him; to have compulsory process for obtaining witnesses in his favor, and to have the assistance of counsel for his defense."
The 6th amendment is additional stuff for the fifth. It goes into more detail about what services must be provided for criminal cases, such as access to a lawyer. Similar to the 5th, it's less relevant than some other amendments but it's still important.
"In suits at common law, where the value in controversy shall exceed twenty dollars, the right of trial by jury shall be preserved, and no fact tried by a jury, shall be otherwise reexamined in any court of the United States, than according to the rules of the common law."
Basically, right to a jury and you can't be charged twice with the same crime. In almost all cases this helps protect people's innocence (there are very few cases where incriminating evidence for a crime was found later and the criminal could not be charged, but this is extremely rare). A very important amendment.
"Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted."
Punishments must suit the crime, basically. You can't be lynched for stealing a $5 keychain in Claire's. The execution of this amendment is extremely debatable, since there are many laws that are considered "cruel and unusual punishment", such as minor violations of drug use or animal welfare laws. In addition, some people cite this amendment when regarding unlawful treatment of people from officers, such as random beatings or tasering.
"The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people."
Basically, just because the rights aren't outlined here in the constitution doesn't mean that they don't exist. Going back to "freedom of religion", this is the counterargument to the "religion vs oppression" debate. For example, you could argue that homosexual marriage should be illegal because it goes against religious rights; however, the other side could argue that due to the 9th amendment, eliminating rights for other people just because it's not in the constitution is unconstitutional. This is often paired with the 13th amendment.
"The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people."
Basically, if it's not mentioned in the constitution, it's a state or people based issue. Not really followed considering many federal laws are not directly in the constitution, but the constitution is pretty clearly not meant to be read literally, considering its ability to be amended.