Hajimemashite means "How do you do?" or "Nice to meet you." This is the very first phrase you say when meeting someone new for the first time. It literally means "For the first time." It's always used as an introduction. For example, "はじめまして。田中と申します。" (How do you do? My name is Tanaka.).

初(はじ)めて人(ひと)に会(あ)う時(とき)に言(い)う挨拶(あいさつ)です。自己(じこ)紹介(しょうかい)の最初(さいしょ)に使(つか)います。「はじめまして！」