It is popularly believed and taught in the Gentile world that the Ten Commandments/Decalogue (Exodus 20:1-17; Deuteronomy 5:5b-21) should be kept/observed by all. For example, the Council of Trent in the Reformation made them binding upon Roman Catholics with this exception: the fourth commndment concerning the Shabbat was changed into a command to observe Sunday, the first day of the week ('the Lord's day'). Is the Decalogue for Jews and Gentiles, or only for Jews?
Non-Jews are held responsible by seven laws, known as the Seven Laws of Noah. All of those laws are contained within the Decalogue, with exception to the third law1, the seventh law2, and to an extent, the fifth law3.
The Decalogue contains several laws not found in the Seven Laws of Noah, namely the fourth command4, the fifth command5, the ninth command6, and the tenth command7.
Clearly the Decalogue was written to be legislation for the Jewish people, based on the inclusion of the fourth law, which is to keep the Sabbath day. Maimonides writes that a non-Jew is forbidden from keeping the Sabbath day, and this idea also has roots in the Talmud (Sanhedrin 58b).
One could ask whether a non-Jew is obligated to honor his parents, etc. even though this is not explicitly written in the Seven Laws of Noah, but that is a separate question entirely.
TL;DR: The Decalogue is for the Jewish people to follow, non-Jews follow a different set of commands known as the Seven Laws of Noah.
1. "To establish courts of justice." 2. "Not to eat flesh torn from a living animal." 3. "Not to commit adultery, bestiality, or sexual immorality." The reason I say "to an extent" is although the prohibition of committing adultery is laid out in the seventh command of the Decalogue, technically that command makes no mention of other sexual immorality. 4. "Remember the Sabbath day." 5. "Honor your father and your mother." 6. "Do not bear false testimony." 7. "Do not covet."