The aseres ha'dibros were engraved by Hakodosh Boruch Hu on two tablets of stone. These two tablets are referred to as the shnei luchos ha'bris - the two stones of the covenant. The reference to a bris has a double connotation:
1) The Vilna Gaon is quoted as having said that the stones themselves represented a gift given by Hakodosh Boruch Hu to the Jewish people, similar to a love bracelet or an engagement ring. When a boy will not be in the presence of the girl he loves, he wants her to have something tangible that should represent his affection towards her.
2) The mitzvos singled out to appear on the luchos are considered, in a certain sense, more important than all the other 613 mitzvos. The mitzvah of milah is considered by the Talmud to be more important than most other mitzvos because it has a bris connected to it (Yevomos 5b). Similarly, all of the mitzvos that appear on the aseres hadibros are more important because they all have a special bris over and above the general bris that is connected with all mitzvos.
In the Shacharis shemoneh esrei of Shabbos, we the highlight the significance of the mitzvah of Shabbos by mentioning its appearance in the aseres ha'dibros and then we add, "v'chein kosuv b'sorosecha", i.e. that the mitzvah of Shabbos appears in another place in the chumash, "v'shomru Bnei Yisroel es haShabbos". Given that the mitzvah of Shabbos appears so many times in the chumash, why did the composers of this brocho select these two sources above all of the others? The explanation given by Rav Chaim Soloveitchick is that all of the mitzvos that are singled out in the aseres ha'dibros had a special bris connected with them, and in the psukim of "v'shomru Bnei Yisroel es haShabbos..." we mention "bris olam", i.e. that Shabbos is different from most other mitzvos in that it was singled out to have its own bris.
The Ramban (in his commentary on the chumash) accepts the opinion of the tannaim which states that the aseres ha'dibros were proclaimed prior to the conversion of the Jewish people, and that they are merely a random sampling of the 613 mitzvos. The presentation of a sampling of mitzvos to the Jewish people before they converted is consistent with the halacha which requires (Yevomos 47b) that before a non-Jew converts, the beis din presiding over his conversion should make him aware of a sampling of the more difficult mitzvos as well as a sampling of the easier mitzvos. Rashi (in his commentary on the chumash) has adopted the opposing position of the tannaim which states that the proclamation of the aseres ha'dibros took place only after the Jewish people had already completed their conversion. Accordingly, Rashi sheds light on the significance of the aseres ha'dibros by quoting Rav Sa'adya Gaon's understanding that the aseres ha'dibros are the ten basic categories of mitzvos under which all of the other 613 mitzvos can be subsumed.
Reb Chaim Soloveitchick's point, that the mitzvos singled out in the aseres ha'dibros have a special status of bris similar to the mitzvah of milah, is obviously in accordance with Rashi's view as opposed to that of the Ramban.
Following Rav Sa'adya Gaon, the basic principles of the Torah are contained in the aseres ha'dibros, and the ten basic principles of nature are the asoroh ma'amoros shebohem nivra ha'olam (Avos 5:1). In olam hazeh we witness contradictions between the moral, ethical, and religious realities on the one hand and the laws of nature on the other hand; we often witness tzaddik v'rah lo and rosha v'tov lo. Le'asid lavoh we will all understand that in fact there was and is no contradiction, rather Hakodosh Boruch Hu always did and does what is really l'tova - for the ultimate best. In this world we often recite the brocho of Dayan ha'emes when a tragedy occurs because while our perception is that we witness a tragedy, we are expected to have emunah that all was really l'tova (see Rabbeinu Yona to Brochos 44b in the pages of the Rif). But the Talmud says (Pesachim 50a) that le'asid lavoh we will never recite Dayan ha'emes because even our perception will be that everything was really l'tova.
One of the great chassidishe rebbes once explained that in the chapter in Tehillim (92) that we recite on Shabbos, "mizmor shir l'yom ha'Shabbos", which according to tradition is a mizmor shir l'osid lovo, we speak about praising Hakodosh Boruch Hu accompanied with music from a ten string harp. Those ten strings represent the idea that le'asid lavoh there will be a harmonious blend between the asorah ma'amoros of the laws of nature and the aseres ha'dibros. Le'asid lavoh we will all realize that there never really was any contradiction.