The ainoi or praises are the rough Byzantine equivalent to the Western Rite service of Lauds, though instead of being a stand-alone service, they are a portion of the end of Orthros, the primary morning office. The term ainoi comes from the repeated "praise" verses in Psalms 148-150 which make up the bulk of the ainoi.
On weekdays, the ainoi are read without stichera inserted between the verses, but on Sundays, resurrectional hymns from the anastasimatarion are interpolated. A common parish practice on Sundays is to sing only the opening verses ("Let everything that hath breath praise the Lord," etc.), and then skip to the verses with the inserted stichera.