In order for a Beracha to be valid, it must consist of "Shem U’malchut" – meaning, it must include G-d’s Name, and it must include a reference to G-d’s Kingship over the world. This is why we say in every Beracha the formula, "Hashem Elokenu Melech Ha’olam." We say "Hashem" – the Name of "Havaya," which we pronounce "Ado-nai" – to fulfill the requirement to mention G-d’s Name, and we say "Melech Ha’olam" ("King of the world") to fulfill the requirement to mention G-d’s Kingship.
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) writes (in Siman 214) that if one used a different Name of G-d, instead of "Havaya," he nevertheless fulfills his requirement. As long as one uses one of the seven Names that we are forbidden to erase ("Sheba Shemot She’enam Nimhakim"), his Beracha is valid. Thus, for example, if one mistakenly recited, "Baruch Ata Sha-ddai," his Beracha is valid and does not need to be repeated. (This applies also if one said "Baruch Ata Ado-nai" but had in mind the Name of "Adnut" instead of the Name of "Havaya"; since "Adnut" is also among the seven Names, the Beracha is valid.)
The Bet Yosef (commentary to the Tur by Maran, author of the Shulhan Aruch) raises the famous question of why the blessings of the Amida prayer do not include "Malchut" – reference to G-d’s Kingship. The blessings recited in the Amida include the phrase "Baruch Ata Hashem," but do not include "Melech Ha’olam." How are these blessings valid, if they do not make mention of G-d’s Kingship?
Citing the Rosh (Rabbenu Asher Ben Yehiel, Germany-Spain, 1250-1327), the Bet Yosef answers that in the first blessing of the Amida, we speak of G-d as "Ha’Kel Ha’gadol Ha’gibor Ve’ha’nora" – "The great, mighty, awesome G-d," which qualifies as a reference to "Malchut." Since this phrase speaks of G-d’s greatness and power, it essentially describes G-d’s reign over the world, thus satisfying the requirement of "Malchut." All the subsequent blessings of the Amida are covered by this reference to "Malchut" in the first blessing, and so they do not require the phrase "Melech Ha’olam." For the same reason, the subsequent blessings of the Amida do not begin with the word "Baruch." Normally, a Beracha must begin with the word "Baruch," but a "Beracha Ha’semucha La’havertah" – a Beracha recited immediately after a different Beracha – is covered by the word "Baruch" that opens the preceding Beracha, and thus does not need to begin with "Baruch." All the Berachot of the Amida prayer are viewed as connected to the first Beracha, and so they are covered both by the word "Baruch" at the beginning of the Amida, as well as by the phrase "Ha’Kel Ha’gadol…" which fulfills the requirement of "Malchut."
The practical Halacha that emerges from this discussion is that one should try to have in mind when reciting the first Beracha of the Amida that the phrase "Ha’Kel Ha’gadol Ha’gibor Ve’ha’nora" is a reference to G-d’s Kingship over the world, in lieu of the phrase "Melech Ha’olam."
Summary: A Beracha must include the Name of G-d, as well as a reference to "Malchut" – His Kingship over the world, which is why we include the phrase "Melech Ha’olam" in every Beracha. The exception is the Amida prayer, in which the requirement of "Malchut" is fulfilled through the recitation of the phrase "Ha’Kel Ha’gadol Ha’gibor Ve’ha’nora" ("The great, mighty, awesome G-d"). Therefore, when one recites this phrase, it is proper to have in mind that he is speaking of G-d’s rule over the universe.