The principle of "Ein Ma'avirin Al Ha'mitzvot" means that one should not pass by an opportunity to perform a Mitzva, and that if a person has several Mitzvot to perform, he should perform the one that he confronts first.
The Shulchan Aruch Ha'Rav (code of law by Rabbi Shneur Zalman of Liadi, the first Rabbi of Lubavitch, Russia, 1745-1813), in Siman 432 (8), applies this rule to the Mitzva of Bedikat Chametz (inspecting one's house for Chametz on the night before Erev Pesach). He writes that a person should first check the rooms nearest to him when he begins the search, rather than pass them by to first search in the farther rooms. The rule of "Ein Ma'avirin Al Ha'mitzvot" requires that one first fulfill the Mitzvot that immediately present themselves, and thus one must first search the rooms he encounters first.
"Ein Ma'avirin Al Ha'mitzvot" also means that one should not unnecessarily delay the performance of a Mitzva. For example, the father of a firstborn son should perform the Mitzva of Pidyon Ha'ben (the symbolic "redemption" of the child by paying money to a Kohen) immediately after the child's thirtieth day, without delay. In fact, the Chida (Rabbi Chayim Yosef David Azulai, Israel, 1724-1806), in his work of responsa Chayim Sha'al (1:31), writes that even if an infant is sick and unable to undergo (brit Milah) circumcision, the Pidyon must not be delayed; it should be performed immediately once the obligation takes effect on the thirty-first day, even if it is before the Brit Milah.
Likewise, one should begin building his Sukka immediately following Yom Kippur, rather than delaying it until one of the days just prior to Sukkot.
Finally, the Ben Ish Chai (Rabbi Yosef Chayim of Baghdad, 1833-1909), in his work of responsa Torah Lishma (66), addresses the situation of a person who left the restroom and, before he recited Asher Yatzar (the Beracha recited after using the restroom), he ate a food requiring a Beracha Acharona. Does he first recite the Beracha of Asher Yatzar, or, once he already delayed the recitation of Asher Yatzar, should he delay it a bit further until after he recites the Beracha Acharona for the food?
The Ben Ish Chai rules that since this individual first came upon the obligation to recite Asher Yatzar, he should first recite this Beracha. The rule of Ein Ma'avirin Al Ha'mitzvot requires that he afford precedence to the Mitzva he encountered first, and therefore in this case one first recites Asher Yatzar, and only then the Beracha Acharona.
Summary: When conducting Bedikat Chametz, one must first check the rooms that he encounters first, rather than skipping them and first inspecting the more distant rooms. A father of a male firstborn should conduct the Pidyon Ha'ben immediately on the infant's thirty-first day, even if the child cannot undergo circumcision at that point. The most opportune time to begin building the Sukka is immediately following Yom Kippur. If a person leaves the restroom and eats something before reciting Asher Yatzar, he first recites Asher Yatzar and only then recites the Beracha Acharona for the food he ate, since Asher Yatzar is the obligation that he had encountered first.