What items are not allowed to be brought aboard space exploration missions?
What can't be brought back from a space mission (without prior authorization)?
Why were these regulations originally put into place?
Disclaimer: IANASL - I Am Not A Space Lawyer :)
This would greatly depend on individual nations carrying out space missions and their national space laws, which would be by an order of magnitude too broad to be reasonably covered within this Q&A, but I presume you're limiting your question to international-law, as the tag you chose implies, and what international laws and regulations are there limiting or prohibiting use of, placement and removal of certain items?
Perhaps most historically important, the Outer Space Treaty of 1967 prohibits placement and use of nuclear weapons or any other weapons of mass destruction in Earth's orbit and beyond (celestial bodies, outer space,...). Additionally, it prohibits building military installations, weapons tests, and similar militarisation of space in it's Article IV:
States Parties to the Treaty undertake not to place in orbit around the earth any objects carrying nuclear weapons or any other kinds of weapons of mass destruction, install such weapons on celestial bodies, or station such weapons in outer space in any other manner. The moon and other celestial bodies shall be used by all States Parties to the Treaty exclusively for peaceful purposes. The establishment of military bases, installations and fortifications, the testing of any type of weapons and the conduct of military manoeuvres on celestial bodies shall be forbidden.
This is later extended to any "potentially harmful interference with activities of other States Parties in the peaceful exploration and use of outer space" in the Article IX of the treaty, but is non-specific regarding any particular items / equipment that is supposed to cause this harmful interference. Most of these "space laws" read as generally agreed upon principles regarding space exploration and use, but aren't specifically mentioning prohibited items.
This said, I personally wouldn't know of any other explicitly prohibited items, but the International Space Law is constantly evolving under the umbrella of United Nations Office for Outer Space Affairs (UNOOSA) in form of international treaties, resolutions, and principles, so I suggest UNOOSA webpage as your first source of information. Naturally, I would be curious to read findings of any other answerers on this matter just as well, and my answer should by no means be considered as complete. Hope it helps, perhaps as a starting point, and others can shed a bit more light on the subject at hand.
This would differ from country to country.
As for the items not allowed to be brought into onto shuttle programs, the items on board the shuttle are very mission specific and rarely are they allowed to carry personal items on board the shuttle.
Like the Space Shuttle program had a Personal Preference Kit or PPK for each crew member.
The contents of a PPK were limited to 20 separate items, with a total weight of 1.5 pounds and all of these items would be placed in a bag provided by NASA.
Soyuz has a little looser allowances in this matter, each crew member could take up to 2.2 pounds of material.
Separate from the PPK are crew care packages. These are manifested by the psychological support teams and include personal items considered to be for the well being of the crew members, such as books, CDs, religious supplies, holiday decorations, and favorite condiments.
In addition, some personal items can be manifested under necessary supplies like, a crew member can have their own clothes included as part of their clothing allowance.
And these packages are checked and once cleared off by NASA's staff. Where they check for any breakable objects. Such as pencils(hence people use special pens in space). or weapons of any sort(That weigh less than 1.5 Pounds).
This weight limit that the Operating bodies have, makes it difficult or rather impossible for them to carry any sort of weapons on board. So the only real danger from the things brought on board by crew members are breakable things which are avoided. Especially because of the lack of gravity, things floating around can hurt crew members
Most space launches are conducted by sovereign states, so there really isn't any authority to prohibit anything. There are some private organizations that launch, but these are generally done under the flag of a sovereign state. In the future, private organizations may choose to operate independently from international waters (ala SeaLaunch), but there is a lack of precedent there.
Having said that, there are some things that receive more scrutiny than others. Here are some to consider: